Calculus Examples

$lim_{x \to 1} \frac{x^{3} - x^{2} - x + 1}{x \sqrt{x} + 1 - \sqrt{x} - x}$

Step 1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 1} x^{3} - x^{2} - x + 1}{lim_{x \to 1} x \sqrt{x} + 1 - \sqrt{x} - x}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{lim_{x \to 1} x^{3} - lim_{x \to 1} x^{2} - lim_{x \to 1} x + lim_{x \to 1} 1}{lim_{x \to 1} x \sqrt{x} + 1 - \sqrt{x} - x}$

Step 1.2.2

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 1} x)}^{3} - lim_{x \to 1} x^{2} - lim_{x \to 1} x + lim_{x \to 1} 1}{lim_{x \to 1} x \sqrt{x} + 1 - \sqrt{x} - x}$

Step 1.2.3

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 1} x)}^{3} - {(lim_{x \to 1} x)}^{2} - lim_{x \to 1} x + lim_{x \to 1} 1}{lim_{x \to 1} x \sqrt{x} + 1 - \sqrt{x} - x}$

Step 1.2.4

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{{(lim_{x \to 1} x)}^{3} - {(lim_{x \to 1} x)}^{2} - lim_{x \to 1} x + 1}{lim_{x \to 1} x \sqrt{x} + 1 - \sqrt{x} - x}$

Step 1.2.5

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 1.2.5.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{1^{3} - {(lim_{x \to 1} x)}^{2} - lim_{x \to 1} x + 1}{lim_{x \to 1} x \sqrt{x} + 1 - \sqrt{x} - x}$

Step 1.2.5.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{1^{3} - 1^{2} - lim_{x \to 1} x + 1}{lim_{x \to 1} x \sqrt{x} + 1 - \sqrt{x} - x}$

Step 1.2.5.3

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{1^{3} - 1^{2} - 1 + 1}{lim_{x \to 1} x \sqrt{x} + 1 - \sqrt{x} - x}$

Step 1.2.6

Simplify the answer.

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Step 1.2.6.1

Simplify each term.

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Step 1.2.6.1.1

One to any power is one.

$\frac{1 - 1^{2} - 1 + 1}{lim_{x \to 1} x \sqrt{x} + 1 - \sqrt{x} - x}$

Step 1.2.6.1.2

One to any power is one.

$\frac{1 - 1 \cdot 1 - 1 + 1}{lim_{x \to 1} x \sqrt{x} + 1 - \sqrt{x} - x}$

Step 1.2.6.1.3

Multiply $- 1$ by $1$ .

$\frac{1 - 1 - 1 + 1}{lim_{x \to 1} x \sqrt{x} + 1 - \sqrt{x} - x}$

Step 1.2.6.2

Subtract $1$ from $1$ .

$\frac{0 - 1 + 1}{lim_{x \to 1} x \sqrt{x} + 1 - \sqrt{x} - x}$

Step 1.2.6.3

Subtract $1$ from $0$ .

$\frac{- 1 + 1}{lim_{x \to 1} x \sqrt{x} + 1 - \sqrt{x} - x}$

Step 1.2.6.4

Add $- 1$ and $1$ .

$\frac{0}{lim_{x \to 1} x \sqrt{x} + 1 - \sqrt{x} - x}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{0}{lim_{x \to 1} x \sqrt{x} + lim_{x \to 1} 1 - lim_{x \to 1} \sqrt{x} - lim_{x \to 1} x}$

Step 1.3.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{0}{lim_{x \to 1} x \cdot lim_{x \to 1} \sqrt{x} + lim_{x \to 1} 1 - lim_{x \to 1} \sqrt{x} - lim_{x \to 1} x}$

Step 1.3.3

Move the limit under the radical sign.

$\frac{0}{lim_{x \to 1} x \cdot \sqrt{lim_{x \to 1} x} + lim_{x \to 1} 1 - lim_{x \to 1} \sqrt{x} - lim_{x \to 1} x}$

Step 1.3.4

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{0}{lim_{x \to 1} x \cdot \sqrt{lim_{x \to 1} x} + 1 - lim_{x \to 1} \sqrt{x} - lim_{x \to 1} x}$

Step 1.3.5

Move the limit under the radical sign.

$\frac{0}{lim_{x \to 1} x \cdot \sqrt{lim_{x \to 1} x} + 1 - \sqrt{lim_{x \to 1} x} - lim_{x \to 1} x}$

Step 1.3.6

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 1.3.6.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{0}{1 \cdot \sqrt{lim_{x \to 1} x} + 1 - \sqrt{lim_{x \to 1} x} - lim_{x \to 1} x}$

Step 1.3.6.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{0}{1 \cdot \sqrt{1} + 1 - \sqrt{lim_{x \to 1} x} - lim_{x \to 1} x}$

Step 1.3.6.3

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{0}{1 \cdot \sqrt{1} + 1 - \sqrt{1} - lim_{x \to 1} x}$

Step 1.3.6.4

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{0}{1 \cdot \sqrt{1} + 1 - \sqrt{1} - 1}$

Step 1.3.7

Simplify the answer.

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Step 1.3.7.1

Simplify each term.

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Step 1.3.7.1.1

Multiply $\sqrt{1}$ by $1$ .

$\frac{0}{\sqrt{1} + 1 - \sqrt{1} - 1}$

Step 1.3.7.1.2

Any root of $1$ is $1$ .

$\frac{0}{1 + 1 - \sqrt{1} - 1}$

Step 1.3.7.1.3

Any root of $1$ is $1$ .

$\frac{0}{1 + 1 - 1 \cdot 1 - 1}$

Step 1.3.7.1.4

Multiply $- 1$ by $1$ .

$\frac{0}{1 + 1 - 1 - 1}$

Step 1.3.7.2

Add $1$ and $1$ .

$\frac{0}{2 - 1 - 1}$

Step 1.3.7.3

Subtract $1$ from $2$ .

$\frac{0}{1 - 1}$

Step 1.3.7.4

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 1.3.7.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.3.8

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 1} \frac{x^{3} - x^{2} - x + 1}{x \sqrt{x} + 1 - \sqrt{x} - x} = lim_{x \to 1} \frac{\frac{d}{d x} [x^{3} - x^{2} - x + 1]}{\frac{d}{d x} [x \sqrt{x} + 1 - \sqrt{x} - x]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 1} \frac{\frac{d}{d x} [x^{3} - x^{2} - x + 1]}{\frac{d}{d x} [x \sqrt{x} + 1 - \sqrt{x} - x]}$

Step 3.2

By the Sum Rule, the derivative of $x^{3} - x^{2} - x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [1]$ .

$lim_{x \to 1} \frac{\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [1]}{\frac{d}{d x} [x \sqrt{x} + 1 - \sqrt{x} - x]}$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$lim_{x \to 1} \frac{3 x^{2} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [1]}{\frac{d}{d x} [x \sqrt{x} + 1 - \sqrt{x} - x]}$

Step 3.4

Evaluate $\frac{d}{d x} [- x^{2}]$ .

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Step 3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$lim_{x \to 1} \frac{3 x^{2} - \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [1]}{\frac{d}{d x} [x \sqrt{x} + 1 - \sqrt{x} - x]}$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 1} \frac{3 x^{2} - (2 x) + \frac{d}{d x} [- x] + \frac{d}{d x} [1]}{\frac{d}{d x} [x \sqrt{x} + 1 - \sqrt{x} - x]}$

Step 3.4.3

Multiply $2$ by $- 1$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x + \frac{d}{d x} [- x] + \frac{d}{d x} [1]}{\frac{d}{d x} [x \sqrt{x} + 1 - \sqrt{x} - x]}$

Step 3.5

Evaluate $\frac{d}{d x} [- x]$ .

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Step 3.5.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - \frac{d}{d x} [x] + \frac{d}{d x} [1]}{\frac{d}{d x} [x \sqrt{x} + 1 - \sqrt{x} - x]}$

Step 3.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1 \cdot 1 + \frac{d}{d x} [1]}{\frac{d}{d x} [x \sqrt{x} + 1 - \sqrt{x} - x]}$

Step 3.5.3

Multiply $- 1$ by $1$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1 + \frac{d}{d x} [1]}{\frac{d}{d x} [x \sqrt{x} + 1 - \sqrt{x} - x]}$

Step 3.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1 + 0}{\frac{d}{d x} [x \sqrt{x} + 1 - \sqrt{x} - x]}$

Step 3.7

Add $3 x^{2} - 2 x - 1$ and $0$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{d}{d x} [x \sqrt{x} + 1 - \sqrt{x} - x]}$

Step 3.8

By the Sum Rule, the derivative of $x \sqrt{x} + 1 - \sqrt{x} - x$ with respect to $x$ is $\frac{d}{d x} [x \sqrt{x}] + \frac{d}{d x} [1] + \frac{d}{d x} [- \sqrt{x}] + \frac{d}{d x} [- x]$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{d}{d x} [x \sqrt{x}] + \frac{d}{d x} [1] + \frac{d}{d x} [- \sqrt{x}] + \frac{d}{d x} [- x]}$

Step 3.9

Evaluate $\frac{d}{d x} [x \sqrt{x}]$ .

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Step 3.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{d}{d x} [x \cdot x^{\frac{1}{2}}] + \frac{d}{d x} [1] + \frac{d}{d x} [- \sqrt{x}] + \frac{d}{d x} [- x]}$

Step 3.9.2

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 3.9.2.1

Multiply $x$ by $x^{\frac{1}{2}}$ .

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Step 3.9.2.1.1

Raise $x$ to the power of $1$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{d}{d x} [x^{1} x^{\frac{1}{2}}] + \frac{d}{d x} [1] + \frac{d}{d x} [- \sqrt{x}] + \frac{d}{d x} [- x]}$

Step 3.9.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{d}{d x} [x^{1 + \frac{1}{2}}] + \frac{d}{d x} [1] + \frac{d}{d x} [- \sqrt{x}] + \frac{d}{d x} [- x]}$

Step 3.9.2.2

Write $1$ as a fraction with a common denominator.

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{d}{d x} [x^{\frac{2}{2} + \frac{1}{2}}] + \frac{d}{d x} [1] + \frac{d}{d x} [- \sqrt{x}] + \frac{d}{d x} [- x]}$

Step 3.9.2.3

Combine the numerators over the common denominator.

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{d}{d x} [x^{\frac{2 + 1}{2}}] + \frac{d}{d x} [1] + \frac{d}{d x} [- \sqrt{x}] + \frac{d}{d x} [- x]}$

Step 3.9.2.4

Add $2$ and $1$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{d}{d x} [x^{\frac{3}{2}}] + \frac{d}{d x} [1] + \frac{d}{d x} [- \sqrt{x}] + \frac{d}{d x} [- x]}$

Step 3.9.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{3}{2} - 1} + \frac{d}{d x} [1] + \frac{d}{d x} [- \sqrt{x}] + \frac{d}{d x} [- x]}$

Step 3.9.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [1] + \frac{d}{d x} [- \sqrt{x}] + \frac{d}{d x} [- x]}$

Step 3.9.5

Combine $- 1$ and $\frac{2}{2}$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [1] + \frac{d}{d x} [- \sqrt{x}] + \frac{d}{d x} [- x]}$

Step 3.9.6

Combine the numerators over the common denominator.

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}} + \frac{d}{d x} [1] + \frac{d}{d x} [- \sqrt{x}] + \frac{d}{d x} [- x]}$

Step 3.9.7

Simplify the numerator.

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Step 3.9.7.1

Multiply $- 1$ by $2$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{3 - 2}{2}} + \frac{d}{d x} [1] + \frac{d}{d x} [- \sqrt{x}] + \frac{d}{d x} [- x]}$

Step 3.9.7.2

Subtract $2$ from $3$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{1}{2}} + \frac{d}{d x} [1] + \frac{d}{d x} [- \sqrt{x}] + \frac{d}{d x} [- x]}$

Step 3.10

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{1}{2}} + 0 + \frac{d}{d x} [- \sqrt{x}] + \frac{d}{d x} [- x]}$

Step 3.11

Evaluate $\frac{d}{d x} [- \sqrt{x}]$ .

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Step 3.11.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{1}{2}} + 0 + \frac{d}{d x} [- x^{\frac{1}{2}}] + \frac{d}{d x} [- x]}$

Step 3.11.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{1}{2}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{1}{2}} + 0 - \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- x]}$

Step 3.11.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{1}{2}} + 0 - (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [- x]}$

Step 3.11.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{1}{2}} + 0 - (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- x]}$

Step 3.11.5

Combine $- 1$ and $\frac{2}{2}$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{1}{2}} + 0 - (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- x]}$

Step 3.11.6

Combine the numerators over the common denominator.

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{1}{2}} + 0 - (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- x]}$

Step 3.11.7

Simplify the numerator.

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Step 3.11.7.1

Multiply $- 1$ by $2$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{1}{2}} + 0 - (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [- x]}$

Step 3.11.7.2

Subtract $2$ from $1$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{1}{2}} + 0 - (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [- x]}$

Step 3.11.8

Move the negative in front of the fraction.

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{1}{2}} + 0 - (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [- x]}$

Step 3.11.9

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{1}{2}} + 0 - \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- x]}$

Step 3.11.10

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{1}{2}} + 0 - \frac{1}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- x]}$

Step 3.12

Evaluate $\frac{d}{d x} [- x]$ .

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Step 3.12.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{1}{2}} + 0 - \frac{1}{2 x^{\frac{1}{2}}} - \frac{d}{d x} [x]}$

Step 3.12.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{1}{2}} + 0 - \frac{1}{2 x^{\frac{1}{2}}} - 1 \cdot 1}$

Step 3.12.3

Multiply $- 1$ by $1$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{1}{2}} + 0 - \frac{1}{2 x^{\frac{1}{2}}} - 1}$

Step 3.13

Simplify.

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Step 3.13.1

Add $\frac{3}{2} x^{\frac{1}{2}}$ and $0$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{1}{2}} - \frac{1}{2 x^{\frac{1}{2}}} - 1}$

Step 3.13.2

Reorder terms.

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3}{2} x^{\frac{1}{2}} - 1 - \frac{1}{2 x^{\frac{1}{2}}}}$

Step 3.13.3

Combine $\frac{3}{2}$ and $x^{\frac{1}{2}}$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3 x^{\frac{1}{2}}}{2} - 1 - \frac{1}{2 x^{\frac{1}{2}}}}$

Step 4

Convert fractional exponents to radicals.

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Step 4.1

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3 \sqrt{x}}{2} - 1 - \frac{1}{2 x^{\frac{1}{2}}}}$

Step 4.2

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3 \sqrt{x}}{2} - 1 - \frac{1}{2 \sqrt{x}}}$

Step 5

Combine terms.

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Step 5.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3 \sqrt{x}}{2} - 1 \cdot \frac{2}{2} - \frac{1}{2 \sqrt{x}}}$

Step 5.2

Combine $- 1$ and $\frac{2}{2}$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3 \sqrt{x}}{2} + \frac{- 1 \cdot 2}{2} - \frac{1}{2 \sqrt{x}}}$

Step 5.3

Combine the numerators over the common denominator.

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3 \sqrt{x} - 1 \cdot 2}{2} - \frac{1}{2 \sqrt{x}}}$

Step 5.4

To write $\frac{3 \sqrt{x} - 1 \cdot 2}{2}$ as a fraction with a common denominator, multiply by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{3 \sqrt{x} - 1 \cdot 2}{2} \cdot \frac{\sqrt{x}}{\sqrt{x}} - \frac{1}{2 \sqrt{x}}}$

Step 5.5

Multiply $\frac{3 \sqrt{x} - 1 \cdot 2}{2}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{(3 \sqrt{x} - 1 \cdot 2) \sqrt{x}}{2 \sqrt{x}} - \frac{1}{2 \sqrt{x}}}$

Step 5.6

Combine the numerators over the common denominator.

$lim_{x \to 1} \frac{3 x^{2} - 2 x - 1}{\frac{(3 \sqrt{x} - 1 \cdot 2) \sqrt{x} - 1}{2 \sqrt{x}}}$

Step 6

Evaluate the limit.

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Step 6.1

Simplify the limit argument.

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Step 6.1.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 1} (3 x^{2} - 2 x - 1) \frac{2 \sqrt{x}}{(3 \sqrt{x} - 1 \cdot 2) \sqrt{x} - 1}$

Step 6.1.2

Multiply $- 1$ by $2$ .

$lim_{x \to 1} (3 x^{2} - 2 x - 1) \frac{2 \sqrt{x}}{(3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 6.1.3

Multiply $3 x^{2} - 2 x - 1$ by $\frac{2 \sqrt{x}}{(3 \sqrt{x} - 2) \sqrt{x} - 1}$ .

$lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) (2 \sqrt{x})}{(3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 6.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) (\sqrt{x})}{(3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7

Apply L'Hospital's rule.

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Step 7.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 7.1.1

Take the limit of the numerator and the limit of the denominator.

$2 \frac{lim_{x \to 1} (3 x^{2} - 2 x - 1) (\sqrt{x})}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.2

Evaluate the limit of the numerator.

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Step 7.1.2.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $1$ .

$2 \frac{lim_{x \to 1} 3 x^{2} - 2 x - 1 \cdot lim_{x \to 1} \sqrt{x}}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.2.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$2 \frac{(lim_{x \to 1} 3 x^{2} - lim_{x \to 1} 2 x - lim_{x \to 1} 1) \cdot lim_{x \to 1} \sqrt{x}}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.2.3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$2 \frac{(3 lim_{x \to 1} x^{2} - lim_{x \to 1} 2 x - lim_{x \to 1} 1) \cdot lim_{x \to 1} \sqrt{x}}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.2.4

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$2 \frac{(3 {(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 2 x - lim_{x \to 1} 1) \cdot lim_{x \to 1} \sqrt{x}}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.2.5

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$2 \frac{(3 {(lim_{x \to 1} x)}^{2} - 2 lim_{x \to 1} x - lim_{x \to 1} 1) \cdot lim_{x \to 1} \sqrt{x}}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.2.6

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$2 \frac{(3 {(lim_{x \to 1} x)}^{2} - 2 lim_{x \to 1} x - 1 \cdot 1) \cdot lim_{x \to 1} \sqrt{x}}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.2.7

Move the limit under the radical sign.

$2 \frac{(3 {(lim_{x \to 1} x)}^{2} - 2 lim_{x \to 1} x - 1 \cdot 1) \cdot \sqrt{lim_{x \to 1} x}}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.2.8

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 7.1.2.8.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$2 \frac{(3 \cdot 1^{2} - 2 lim_{x \to 1} x - 1 \cdot 1) \cdot \sqrt{lim_{x \to 1} x}}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.2.8.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$2 \frac{(3 \cdot 1^{2} - 2 \cdot 1 - 1 \cdot 1) \cdot \sqrt{lim_{x \to 1} x}}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.2.8.3

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$2 \frac{(3 \cdot 1^{2} - 2 \cdot 1 - 1 \cdot 1) \cdot \sqrt{1}}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.2.9

Simplify the answer.

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Step 7.1.2.9.1

Simplify each term.

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Step 7.1.2.9.1.1

One to any power is one.

$2 \frac{(3 \cdot 1 - 2 \cdot 1 - 1 \cdot 1) \cdot \sqrt{1}}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.2.9.1.2

Multiply $3$ by $1$ .

$2 \frac{(3 - 2 \cdot 1 - 1 \cdot 1) \cdot \sqrt{1}}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.2.9.1.3

Multiply $- 2$ by $1$ .

$2 \frac{(3 - 2 - 1 \cdot 1) \cdot \sqrt{1}}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.2.9.1.4

Multiply $- 1$ by $1$ .

$2 \frac{(3 - 2 - 1) \cdot \sqrt{1}}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.2.9.2

Subtract $2$ from $3$ .

$2 \frac{(1 - 1) \cdot \sqrt{1}}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.2.9.3

Subtract $1$ from $1$ .

$2 \frac{0 \cdot \sqrt{1}}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.2.9.4

Any root of $1$ is $1$ .

$2 \frac{0 \cdot 1}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.2.9.5

Multiply $0$ by $1$ .

$2 \frac{0}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - 1}$

Step 7.1.3

Evaluate the limit of the denominator.

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Step 7.1.3.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$2 \frac{0}{lim_{x \to 1} (3 \sqrt{x} - 2) \sqrt{x} - lim_{x \to 1} 1}$

Step 7.1.3.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $1$ .

$2 \frac{0}{lim_{x \to 1} 3 \sqrt{x} - 2 \cdot lim_{x \to 1} \sqrt{x} - lim_{x \to 1} 1}$

Step 7.1.3.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$2 \frac{0}{(lim_{x \to 1} 3 \sqrt{x} - lim_{x \to 1} 2) \cdot lim_{x \to 1} \sqrt{x} - lim_{x \to 1} 1}$

Step 7.1.3.4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$2 \frac{0}{(3 lim_{x \to 1} \sqrt{x} - lim_{x \to 1} 2) \cdot lim_{x \to 1} \sqrt{x} - lim_{x \to 1} 1}$

Step 7.1.3.5

Move the limit under the radical sign.

$2 \frac{0}{(3 \sqrt{lim_{x \to 1} x} - lim_{x \to 1} 2) \cdot lim_{x \to 1} \sqrt{x} - lim_{x \to 1} 1}$

Step 7.1.3.6

Evaluate the limit of $2$ which is constant as $x$ approaches $1$ .

$2 \frac{0}{(3 \sqrt{lim_{x \to 1} x} - 1 \cdot 2) \cdot lim_{x \to 1} \sqrt{x} - lim_{x \to 1} 1}$

Step 7.1.3.7

Move the limit under the radical sign.

$2 \frac{0}{(3 \sqrt{lim_{x \to 1} x} - 1 \cdot 2) \cdot \sqrt{lim_{x \to 1} x} - lim_{x \to 1} 1}$

Step 7.1.3.8

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$2 \frac{0}{(3 \sqrt{lim_{x \to 1} x} - 1 \cdot 2) \cdot \sqrt{lim_{x \to 1} x} - 1 \cdot 1}$

Step 7.1.3.9

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 7.1.3.9.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$2 \frac{0}{(3 \sqrt{1} - 1 \cdot 2) \cdot \sqrt{lim_{x \to 1} x} - 1 \cdot 1}$

Step 7.1.3.9.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$2 \frac{0}{(3 \sqrt{1} - 1 \cdot 2) \cdot \sqrt{1} - 1 \cdot 1}$

Step 7.1.3.10

Simplify the answer.

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Step 7.1.3.10.1

Simplify each term.

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Step 7.1.3.10.1.1

Simplify each term.

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Step 7.1.3.10.1.1.1

Any root of $1$ is $1$ .

$2 \frac{0}{(3 \cdot 1 - 1 \cdot 2) \cdot \sqrt{1} - 1 \cdot 1}$

Step 7.1.3.10.1.1.2

Multiply $3$ by $1$ .

$2 \frac{0}{(3 - 1 \cdot 2) \cdot \sqrt{1} - 1 \cdot 1}$

Step 7.1.3.10.1.1.3

Multiply $- 1$ by $2$ .

$2 \frac{0}{(3 - 2) \cdot \sqrt{1} - 1 \cdot 1}$

Step 7.1.3.10.1.2

Subtract $2$ from $3$ .

$2 \frac{0}{1 \cdot \sqrt{1} - 1 \cdot 1}$

Step 7.1.3.10.1.3

Multiply $\sqrt{1}$ by $1$ .

$2 \frac{0}{\sqrt{1} - 1 \cdot 1}$

Step 7.1.3.10.1.4

Any root of $1$ is $1$ .

$2 \frac{0}{1 - 1 \cdot 1}$

Step 7.1.3.10.1.5

Multiply $- 1$ by $1$ .

$2 \frac{0}{1 - 1}$

Step 7.1.3.10.2

Subtract $1$ from $1$ .

$2 (\frac{0}{0})$

Step 7.1.3.10.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$2 (\frac{0}{0})$

Step 7.1.3.11

The expression contains a division by $0$ . The expression is undefined.

Undefined

$2 (\frac{0}{0})$

Step 7.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$2 (\frac{0}{0})$

Step 7.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) (\sqrt{x})}{(3 \sqrt{x} - 2) \sqrt{x} - 1} = lim_{x \to 1} \frac{\frac{d}{d x} [(3 x^{2} - 2 x - 1) (\sqrt{x})]}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3

Find the derivative of the numerator and denominator.

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Step 7.3.1

Differentiate the numerator and denominator.

$2 lim_{x \to 1} \frac{\frac{d}{d x} [(3 x^{2} - 2 x - 1) (\sqrt{x})]}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$2 lim_{x \to 1} \frac{\frac{d}{d x} [(3 x^{2} - 2 x - 1) x^{\frac{1}{2}}]}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 3 x^{2} - 2 x - 1$ and $g (x) = x^{\frac{1}{2}}$ .

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{d}{d x} [x^{\frac{1}{2}}] + x^{\frac{1}{2}} \frac{d}{d x} [3 x^{2} - 2 x - 1]}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{1}{2} x^{\frac{1}{2} - 1} + x^{\frac{1}{2}} \frac{d}{d x} [3 x^{2} - 2 x - 1]}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + x^{\frac{1}{2}} \frac{d}{d x} [3 x^{2} - 2 x - 1]}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.6

Combine $- 1$ and $\frac{2}{2}$ .

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + x^{\frac{1}{2}} \frac{d}{d x} [3 x^{2} - 2 x - 1]}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.7

Combine the numerators over the common denominator.

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + x^{\frac{1}{2}} \frac{d}{d x} [3 x^{2} - 2 x - 1]}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.8

Simplify the numerator.

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Step 7.3.8.1

Multiply $- 1$ by $2$ .

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{1}{2} x^{\frac{1 - 2}{2}} + x^{\frac{1}{2}} \frac{d}{d x} [3 x^{2} - 2 x - 1]}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.8.2

Subtract $2$ from $1$ .

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{1}{2} x^{\frac{- 1}{2}} + x^{\frac{1}{2}} \frac{d}{d x} [3 x^{2} - 2 x - 1]}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.9

Move the negative in front of the fraction.

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{1}{2} x^{- \frac{1}{2}} + x^{\frac{1}{2}} \frac{d}{d x} [3 x^{2} - 2 x - 1]}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.10

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{x^{- \frac{1}{2}}}{2} + x^{\frac{1}{2}} \frac{d}{d x} [3 x^{2} - 2 x - 1]}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.11

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \frac{d}{d x} [3 x^{2} - 2 x - 1]}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.12

By the Sum Rule, the derivative of $3 x^{2} - 2 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 1]$ .

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 1])}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.13

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 1])}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.14

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (3 \cdot 2 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 1])}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.15

Multiply $2$ by $3$ .

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (6 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 1])}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.16

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (6 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.17

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (6 x - 2 \cdot 1 + \frac{d}{d x} [- 1])}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.18

Multiply $- 2$ by $1$ .

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (6 x - 2 + \frac{d}{d x} [- 1])}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.19

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (6 x - 2 + 0)}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.20

Add $6 x - 2$ and $0$ .

$2 lim_{x \to 1} \frac{(3 x^{2} - 2 x - 1) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (6 x - 2)}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21

Simplify.

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Step 7.3.21.1

Apply the distributive property.

$2 lim_{x \to 1} \frac{3 x^{2} \frac{1}{2 x^{\frac{1}{2}}} - 2 x \frac{1}{2 x^{\frac{1}{2}}} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (6 x - 2)}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.2

Apply the distributive property.

$2 lim_{x \to 1} \frac{3 x^{2} \frac{1}{2 x^{\frac{1}{2}}} - 2 x \frac{1}{2 x^{\frac{1}{2}}} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3

Combine terms.

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Step 7.3.21.3.1

Combine $3$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$2 lim_{x \to 1} \frac{x^{2} \frac{3}{2 x^{\frac{1}{2}}} - 2 x \frac{1}{2 x^{\frac{1}{2}}} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.2

Combine $x^{2}$ and $\frac{3}{2 x^{\frac{1}{2}}}$ .

$2 lim_{x \to 1} \frac{\frac{x^{2} \cdot 3}{2 x^{\frac{1}{2}}} - 2 x \frac{1}{2 x^{\frac{1}{2}}} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.3

Move $3$ to the left of $x^{2}$ .

$2 lim_{x \to 1} \frac{\frac{3 \cdot x^{2}}{2 x^{\frac{1}{2}}} - 2 x \frac{1}{2 x^{\frac{1}{2}}} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.4

Move $x^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{2} x^{- \frac{1}{2}}}{2} - 2 x \frac{1}{2 x^{\frac{1}{2}}} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.5

Multiply $x^{2}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 7.3.21.3.5.1

Move $x^{- \frac{1}{2}}$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{- \frac{1}{2}} x^{2}}{2} - 2 x \frac{1}{2 x^{\frac{1}{2}}} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 lim_{x \to 1} \frac{\frac{3 x^{- \frac{1}{2} + 2}}{2} - 2 x \frac{1}{2 x^{\frac{1}{2}}} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.5.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{- \frac{1}{2} + 2 \cdot \frac{2}{2}}}{2} - 2 x \frac{1}{2 x^{\frac{1}{2}}} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.5.4

Combine $2$ and $\frac{2}{2}$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{- \frac{1}{2} + \frac{2 \cdot 2}{2}}}{2} - 2 x \frac{1}{2 x^{\frac{1}{2}}} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.5.5

Combine the numerators over the common denominator.

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{- 1 + 2 \cdot 2}{2}}}{2} - 2 x \frac{1}{2 x^{\frac{1}{2}}} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.5.6

Simplify the numerator.

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Step 7.3.21.3.5.6.1

Multiply $2$ by $2$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{- 1 + 4}{2}}}{2} - 2 x \frac{1}{2 x^{\frac{1}{2}}} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.5.6.2

Add $- 1$ and $4$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} - 2 x \frac{1}{2 x^{\frac{1}{2}}} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.6

Combine $- 2$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} + x \frac{- 2}{2 x^{\frac{1}{2}}} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.7

Combine $x$ and $\frac{- 2}{2 x^{\frac{1}{2}}}$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} + \frac{x \cdot - 2}{2 x^{\frac{1}{2}}} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.8

Move $- 2$ to the left of $x$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} + \frac{- 2 \cdot x}{2 x^{\frac{1}{2}}} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.9

Move $x^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} + \frac{- 2 x \cdot x^{- \frac{1}{2}}}{2} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.10

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 7.3.21.3.10.1

Move $x^{- \frac{1}{2}}$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} + \frac{- 2 x^{- \frac{1}{2}} x}{2} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.10.2

Multiply $x^{- \frac{1}{2}}$ by $x$ .

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Step 7.3.21.3.10.2.1

Raise $x$ to the power of $1$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} + \frac{- 2 x^{- \frac{1}{2}} x^{1}}{2} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} + \frac{- 2 x^{- \frac{1}{2} + 1}}{2} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.10.3

Write $1$ as a fraction with a common denominator.

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} + \frac{- 2 x^{- \frac{1}{2} + \frac{2}{2}}}{2} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.10.4

Combine the numerators over the common denominator.

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} + \frac{- 2 x^{\frac{- 1 + 2}{2}}}{2} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.10.5

Add $- 1$ and $2$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} + \frac{- 2 x^{\frac{1}{2}}}{2} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.11

Factor $2$ out of $- 2 x^{\frac{1}{2}}$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} + \frac{2 \cdot - 1 x^{\frac{1}{2}}}{2} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.12

Cancel the common factors.

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Step 7.3.21.3.12.1

Factor $2$ out of $2$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} + \frac{2 \cdot - 1 x^{\frac{1}{2}}}{2 (1)} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.12.2

Cancel the common factor.

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} + \frac{2 \cdot - 1 x^{\frac{1}{2}}}{2 \cdot 1} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.12.3

Rewrite the expression.

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} + \frac{- x^{\frac{1}{2}}}{1} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.12.4

Divide $- x^{\frac{1}{2}}$ by $1$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} - x^{\frac{1}{2}} - 1 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.13

Rewrite $- 1 \frac{1}{2 x^{\frac{1}{2}}}$ as $- \frac{1}{2 x^{\frac{1}{2}}}$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} - x^{\frac{1}{2}} - \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \cdot 6 x + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.14

Multiply $x^{\frac{1}{2}}$ by $x$ by adding the exponents.

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Step 7.3.21.3.14.1

Move $x$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} - x^{\frac{1}{2}} - \frac{1}{2 x^{\frac{1}{2}}} + x \cdot x^{\frac{1}{2}} \cdot 6 + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.14.2

Multiply $x$ by $x^{\frac{1}{2}}$ .

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Step 7.3.21.3.14.2.1

Raise $x$ to the power of $1$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} - x^{\frac{1}{2}} - \frac{1}{2 x^{\frac{1}{2}}} + x^{1} x^{\frac{1}{2}} \cdot 6 + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.14.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} - x^{\frac{1}{2}} - \frac{1}{2 x^{\frac{1}{2}}} + x^{1 + \frac{1}{2}} \cdot 6 + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.14.3

Write $1$ as a fraction with a common denominator.

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} - x^{\frac{1}{2}} - \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{2}{2} + \frac{1}{2}} \cdot 6 + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.14.4

Combine the numerators over the common denominator.

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} - x^{\frac{1}{2}} - \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{2 + 1}{2}} \cdot 6 + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.14.5

Add $2$ and $1$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} - x^{\frac{1}{2}} - \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{3}{2}} \cdot 6 + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.15

Move $6$ to the left of $x^{\frac{3}{2}}$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} - x^{\frac{1}{2}} - \frac{1}{2 x^{\frac{1}{2}}} + 6 \cdot x^{\frac{3}{2}} + x^{\frac{1}{2}} \cdot - 2}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.16

Move $- 2$ to the left of $x^{\frac{1}{2}}$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} - x^{\frac{1}{2}} - \frac{1}{2 x^{\frac{1}{2}}} + 6 x^{\frac{3}{2}} - 2 \cdot x^{\frac{1}{2}}}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.17

To write $6 x^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} + 6 x^{\frac{3}{2}} \cdot \frac{2}{2} - x^{\frac{1}{2}} - \frac{1}{2 x^{\frac{1}{2}}} - 2 x^{\frac{1}{2}}}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.18

Combine $6 x^{\frac{3}{2}}$ and $\frac{2}{2}$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}}}{2} + \frac{6 x^{\frac{3}{2}} \cdot 2}{2} - x^{\frac{1}{2}} - \frac{1}{2 x^{\frac{1}{2}}} - 2 x^{\frac{1}{2}}}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.19

Combine the numerators over the common denominator.

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}} + 6 x^{\frac{3}{2}} \cdot 2}{2} - x^{\frac{1}{2}} - \frac{1}{2 x^{\frac{1}{2}}} - 2 x^{\frac{1}{2}}}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.20

Multiply $2$ by $6$ .

$2 lim_{x \to 1} \frac{\frac{3 x^{\frac{3}{2}} + 12 x^{\frac{3}{2}}}{2} - x^{\frac{1}{2}} - \frac{1}{2 x^{\frac{1}{2}}} - 2 x^{\frac{1}{2}}}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.21

Add $3 x^{\frac{3}{2}}$ and $12 x^{\frac{3}{2}}$ .

$2 lim_{x \to 1} \frac{\frac{15 x^{\frac{3}{2}}}{2} - x^{\frac{1}{2}} - \frac{1}{2 x^{\frac{1}{2}}} - 2 x^{\frac{1}{2}}}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.3.22

Subtract $2 x^{\frac{1}{2}}$ from $- x^{\frac{1}{2}}$ .

$2 lim_{x \to 1} \frac{\frac{15 x^{\frac{3}{2}}}{2} - 3 x^{\frac{1}{2}} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.21.4

Reorder terms.

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x} - 1]}$

Step 7.3.22

By the Sum Rule, the derivative of $(3 \sqrt{x} - 2) \sqrt{x} - 1$ with respect to $x$ is $\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x}] + \frac{d}{d x} [- 1]$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x}] + \frac{d}{d x} [- 1]}$

Step 7.3.23

Evaluate $\frac{d}{d x} [(3 \sqrt{x} - 2) \sqrt{x}]$ .

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Step 7.3.23.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [(3 x^{\frac{1}{2}} - 2) \sqrt{x}] + \frac{d}{d x} [- 1]}$

Step 7.3.23.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [(3 x^{\frac{1}{2}} - 2) x^{\frac{1}{2}}] + \frac{d}{d x} [- 1]}$

Step 7.3.23.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 3 x^{\frac{1}{2}} - 2$ and $g (x) = x^{\frac{1}{2}}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{d}{d x} [x^{\frac{1}{2}}] + x^{\frac{1}{2}} \frac{d}{d x} [3 x^{\frac{1}{2}} - 2] + \frac{d}{d x} [- 1]}$

Step 7.3.23.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2} x^{\frac{1}{2} - 1} + x^{\frac{1}{2}} \frac{d}{d x} [3 x^{\frac{1}{2}} - 2] + \frac{d}{d x} [- 1]}$

Step 7.3.23.5

By the Sum Rule, the derivative of $3 x^{\frac{1}{2}} - 2$ with respect to $x$ is $\frac{d}{d x} [3 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2} x^{\frac{1}{2} - 1} + x^{\frac{1}{2}} (\frac{d}{d x} [3 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]) + \frac{d}{d x} [- 1]}$

Step 7.3.23.6

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{\frac{1}{2}}$ with respect to $x$ is $3 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2} x^{\frac{1}{2} - 1} + x^{\frac{1}{2}} (3 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]) + \frac{d}{d x} [- 1]}$

Step 7.3.23.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2} x^{\frac{1}{2} - 1} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + \frac{d}{d x} [- 2]) + \frac{d}{d x} [- 1]}$

Step 7.3.23.8

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2} x^{\frac{1}{2} - 1} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [- 1]}$

Step 7.3.23.9

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [- 1]}$

Step 7.3.23.10

Combine $- 1$ and $\frac{2}{2}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [- 1]}$

Step 7.3.23.11

Combine the numerators over the common denominator.

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [- 1]}$

Step 7.3.23.12

Simplify the numerator.

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Step 7.3.23.12.1

Multiply $- 1$ by $2$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2} x^{\frac{1 - 2}{2}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [- 1]}$

Step 7.3.23.12.2

Subtract $2$ from $1$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2} x^{\frac{- 1}{2}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [- 1]}$

Step 7.3.23.13

Move the negative in front of the fraction.

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2} x^{- \frac{1}{2}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [- 1]}$

Step 7.3.23.14

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{x^{- \frac{1}{2}}}{2} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [- 1]}$

Step 7.3.23.15

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [- 1]}$

Step 7.3.23.16

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

Step 7.3.23.17

Combine $- 1$ and $\frac{2}{2}$ .

Step 7.3.23.18

Combine the numerators over the common denominator.

Step 7.3.23.19

Simplify the numerator.

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Step 7.3.23.19.1

Multiply $- 1$ by $2$ .

Step 7.3.23.19.2

Subtract $2$ from $1$ .

Step 7.3.23.20

Move the negative in front of the fraction.

Step 7.3.23.21

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (3 \frac{x^{- \frac{1}{2}}}{2} + 0) + \frac{d}{d x} [- 1]}$

Step 7.3.23.22

Combine $3$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (\frac{3 x^{- \frac{1}{2}}}{2} + 0) + \frac{d}{d x} [- 1]}$

Step 7.3.23.23

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (\frac{3}{2 x^{\frac{1}{2}}} + 0) + \frac{d}{d x} [- 1]}$

Step 7.3.23.24

Add $\frac{3}{2 x^{\frac{1}{2}}}$ and $0$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \frac{3}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 1]}$

Step 7.3.23.25

Combine $x^{\frac{1}{2}}$ and $\frac{3}{2 x^{\frac{1}{2}}}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2 x^{\frac{1}{2}}} + \frac{x^{\frac{1}{2}} \cdot 3}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 1]}$

Step 7.3.23.26

Move $3$ to the left of $x^{\frac{1}{2}}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2 x^{\frac{1}{2}}} + \frac{3 \cdot x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 1]}$

Step 7.3.23.27

Cancel the common factor.

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2 x^{\frac{1}{2}}} + \frac{3 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 1]}$

Step 7.3.23.28

Rewrite the expression.

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2 x^{\frac{1}{2}}} + \frac{3}{2} + \frac{d}{d x} [- 1]}$

Step 7.3.23.29

To write $(3 x^{\frac{1}{2}} - 2) \frac{1}{2 x^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 2) \frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{2}{2} + \frac{3}{2} + \frac{d}{d x} [- 1]}$

Step 7.3.23.30

Combine $(3 x^{\frac{1}{2}} - 2) \frac{1}{2 x^{\frac{1}{2}}}$ and $\frac{2}{2}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{(3 x^{\frac{1}{2}} - 2) \frac{1}{2 x^{\frac{1}{2}}} \cdot 2}{2} + \frac{3}{2} + \frac{d}{d x} [- 1]}$

Step 7.3.23.31

Combine the numerators over the common denominator.

Step 7.3.23.32

Combine $2$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{(3 x^{\frac{1}{2}} - 2) \frac{2}{2 x^{\frac{1}{2}}} + 3}{2} + \frac{d}{d x} [- 1]}$

Step 7.3.23.33

Cancel the common factor.

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{(3 x^{\frac{1}{2}} - 2) \frac{2}{2 x^{\frac{1}{2}}} + 3}{2} + \frac{d}{d x} [- 1]}$

Step 7.3.23.34

Rewrite the expression.

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{(3 x^{\frac{1}{2}} - 2) \frac{1}{x^{\frac{1}{2}}} + 3}{2} + \frac{d}{d x} [- 1]}$

Step 7.3.24

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{(3 x^{\frac{1}{2}} - 2) \frac{1}{x^{\frac{1}{2}}} + 3}{2} + 0}$

Step 7.3.25

Simplify.

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Step 7.3.25.1

Apply the distributive property.

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{3 x^{\frac{1}{2}} \frac{1}{x^{\frac{1}{2}}} - 2 \frac{1}{x^{\frac{1}{2}}} + 3}{2} + 0}$

Step 7.3.25.2

Combine terms.

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Step 7.3.25.2.1

Combine $3$ and $\frac{1}{x^{\frac{1}{2}}}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} - 2 \frac{1}{x^{\frac{1}{2}}} + 3}{2} + 0}$

Step 7.3.25.2.2

Combine $x^{\frac{1}{2}}$ and $\frac{3}{x^{\frac{1}{2}}}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{\frac{x^{\frac{1}{2}} \cdot 3}{x^{\frac{1}{2}}} - 2 \frac{1}{x^{\frac{1}{2}}} + 3}{2} + 0}$

Step 7.3.25.2.3

Move $3$ to the left of $x^{\frac{1}{2}}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{\frac{3 \cdot x^{\frac{1}{2}}}{x^{\frac{1}{2}}} - 2 \frac{1}{x^{\frac{1}{2}}} + 3}{2} + 0}$

Step 7.3.25.2.4

Cancel the common factor.

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{\frac{3 x^{\frac{1}{2}}}{x^{\frac{1}{2}}} - 2 \frac{1}{x^{\frac{1}{2}}} + 3}{2} + 0}$

Step 7.3.25.2.5

Divide $3$ by $1$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{3 - 2 \frac{1}{x^{\frac{1}{2}}} + 3}{2} + 0}$

Step 7.3.25.2.6

Combine $- 2$ and $\frac{1}{x^{\frac{1}{2}}}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{3 + \frac{- 2}{x^{\frac{1}{2}}} + 3}{2} + 0}$

Step 7.3.25.2.7

Move the negative in front of the fraction.

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{3 - \frac{2}{x^{\frac{1}{2}}} + 3}{2} + 0}$

Step 7.3.25.2.8

Add $3$ and $3$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{6 - \frac{2}{x^{\frac{1}{2}}}}{2} + 0}$

Step 7.3.25.2.9

Factor $2$ out of $6$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{2 \cdot 3 - \frac{2}{x^{\frac{1}{2}}}}{2} + 0}$

Step 7.3.25.2.10

Factor $2$ out of $- \frac{2}{x^{\frac{1}{2}}}$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{2 \cdot 3 + 2 \cdot - 1 \frac{1}{x^{\frac{1}{2}}}}{2} + 0}$

Step 7.3.25.2.11

Factor $2$ out of $2 (3) + 2 (- \frac{1}{x^{\frac{1}{2}}})$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{2 (3 - \frac{1}{x^{\frac{1}{2}}})}{2} + 0}$

Step 7.3.25.2.12

Cancel the common factors.

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Step 7.3.25.2.12.1

Factor $2$ out of $2$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{2 (3 - \frac{1}{x^{\frac{1}{2}}})}{2 (1)} + 0}$

Step 7.3.25.2.12.2

Cancel the common factor.

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{2 (3 - \frac{1}{x^{\frac{1}{2}}})}{2 \cdot 1} + 0}$

Step 7.3.25.2.12.3

Rewrite the expression.

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{3 - \frac{1}{x^{\frac{1}{2}}}}{1} + 0}$

Step 7.3.25.2.12.4

Divide $3 - \frac{1}{x^{\frac{1}{2}}}$ by $1$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{3 - \frac{1}{x^{\frac{1}{2}}} + 0}$

Step 7.3.25.2.13

Add $3 - \frac{1}{x^{\frac{1}{2}}}$ and $0$ .

$2 lim_{x \to 1} \frac{- 3 x^{\frac{1}{2}} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{3 - \frac{1}{x^{\frac{1}{2}}}}$

Step 7.4

Convert fractional exponents to radicals.

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Step 7.4.1

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$2 lim_{x \to 1} \frac{- 3 \sqrt{x} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}{3 - \frac{1}{x^{\frac{1}{2}}}}$

Step 7.4.2

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$2 lim_{x \to 1} \frac{- 3 \sqrt{x} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 \sqrt{x}}}{3 - \frac{1}{x^{\frac{1}{2}}}}$

Step 7.4.3

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$2 lim_{x \to 1} \frac{- 3 \sqrt{x} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 \sqrt{x}}}{3 - \frac{1}{\sqrt{x}}}$

Step 7.5

Combine terms.

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Step 7.5.1

To write $- 3 \sqrt{x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 lim_{x \to 1} \frac{- 3 \sqrt{x} \cdot \frac{2}{2} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 \sqrt{x}}}{3 - \frac{1}{\sqrt{x}}}$

Step 7.5.2

Combine $- 3 \sqrt{x}$ and $\frac{2}{2}$ .

$2 lim_{x \to 1} \frac{\frac{- 3 \sqrt{x} \cdot 2}{2} + \frac{15 x^{\frac{3}{2}}}{2} - \frac{1}{2 \sqrt{x}}}{3 - \frac{1}{\sqrt{x}}}$

Step 7.5.3

Combine the numerators over the common denominator.

$2 lim_{x \to 1} \frac{\frac{- 3 \sqrt{x} \cdot 2 + 15 x^{\frac{3}{2}}}{2} - \frac{1}{2 \sqrt{x}}}{3 - \frac{1}{\sqrt{x}}}$

Step 7.5.4

To write $\frac{- 3 \sqrt{x} \cdot 2 + 15 x^{\frac{3}{2}}}{2}$ as a fraction with a common denominator, multiply by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$2 lim_{x \to 1} \frac{\frac{- 3 \sqrt{x} \cdot 2 + 15 x^{\frac{3}{2}}}{2} \cdot \frac{\sqrt{x}}{\sqrt{x}} - \frac{1}{2 \sqrt{x}}}{3 - \frac{1}{\sqrt{x}}}$

Step 7.5.5

Multiply $\frac{- 3 \sqrt{x} \cdot 2 + 15 x^{\frac{3}{2}}}{2}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$2 lim_{x \to 1} \frac{\frac{(- 3 \sqrt{x} \cdot 2 + 15 x^{\frac{3}{2}}) \sqrt{x}}{2 \sqrt{x}} - \frac{1}{2 \sqrt{x}}}{3 - \frac{1}{\sqrt{x}}}$

Step 7.5.6

Combine the numerators over the common denominator.

$2 lim_{x \to 1} \frac{\frac{(- 3 \sqrt{x} \cdot 2 + 15 x^{\frac{3}{2}}) \sqrt{x} - 1}{2 \sqrt{x}}}{3 - \frac{1}{\sqrt{x}}}$

Step 7.5.7

To write $3$ as a fraction with a common denominator, multiply by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$2 lim_{x \to 1} \frac{\frac{(- 3 \sqrt{x} \cdot 2 + 15 x^{\frac{3}{2}}) \sqrt{x} - 1}{2 \sqrt{x}}}{\frac{3 \sqrt{x}}{\sqrt{x}} - \frac{1}{\sqrt{x}}}$

Step 7.5.8

Combine the numerators over the common denominator.

$2 lim_{x \to 1} \frac{\frac{(- 3 \sqrt{x} \cdot 2 + 15 x^{\frac{3}{2}}) \sqrt{x} - 1}{2 \sqrt{x}}}{\frac{3 \sqrt{x} - 1}{\sqrt{x}}}$

Step 8

Evaluate the limit.

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Step 8.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

$2 \frac{lim_{x \to 1} \frac{(- 3 \sqrt{x} \cdot 2 + 15 x^{\frac{3}{2}}) \sqrt{x} - 1}{2 \sqrt{x}}}{lim_{x \to 1} \frac{3 \sqrt{x} - 1}{\sqrt{x}}}$

Step 8.2

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$2 \frac{\frac{1}{2} lim_{x \to 1} \frac{(- 3 \sqrt{x} \cdot 2 + 15 x^{\frac{3}{2}}) \sqrt{x} - 1}{\sqrt{x}}}{lim_{x \to 1} \frac{3 \sqrt{x} - 1}{\sqrt{x}}}$

Step 8.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

$2 \frac{\frac{1}{2} \cdot \frac{lim_{x \to 1} (- 3 \sqrt{x} \cdot 2 + 15 x^{\frac{3}{2}}) \sqrt{x} - 1}{lim_{x \to 1} \sqrt{x}}}{lim_{x \to 1} \frac{3 \sqrt{x} - 1}{\sqrt{x}}}$

Step 8.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$2 \frac{\frac{1}{2} \cdot \frac{lim_{x \to 1} (- 3 \sqrt{x} \cdot 2 + 15 x^{\frac{3}{2}}) \sqrt{x} - lim_{x \to 1} 1}{lim_{x \to 1} \sqrt{x}}}{lim_{x \to 1} \frac{3 \sqrt{x} - 1}{\sqrt{x}}}$

Step 8.5

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $1$ .

$2 \frac{\frac{1}{2} \cdot \frac{lim_{x \to 1} - 3 \sqrt{x} \cdot 2 + 15 x^{\frac{3}{2}} \cdot lim_{x \to 1} \sqrt{x} - lim_{x \to 1} 1}{lim_{x \to 1} \sqrt{x}}}{lim_{x \to 1} \frac{3 \sqrt{x} - 1}{\sqrt{x}}}$

Step 8.6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$2 \frac{\frac{1}{2} \cdot \frac{(- lim_{x \to 1} 3 \sqrt{x} \cdot 2 + lim_{x \to 1} 15 x^{\frac{3}{2}}) \cdot lim_{x \to 1} \sqrt{x} - lim_{x \to 1} 1}{lim_{x \to 1} \sqrt{x}}}{lim_{x \to 1} \frac{3 \sqrt{x} - 1}{\sqrt{x}}}$

Step 8.7

Move the term $3 \cdot 2$ outside of the limit because it is constant with respect to $x$ .

$2 \frac{\frac{1}{2} \cdot \frac{(- 1 \cdot 3 \cdot 2 lim_{x \to 1} \sqrt{x} + lim_{x \to 1} 15 x^{\frac{3}{2}}) \cdot lim_{x \to 1} \sqrt{x} - lim_{x \to 1} 1}{lim_{x \to 1} \sqrt{x}}}{lim_{x \to 1} \frac{3 \sqrt{x} - 1}{\sqrt{x}}}$

Step 8.8

Move the limit under the radical sign.

$2 \frac{\frac{1}{2} \cdot \frac{(- 1 \cdot 3 \cdot 2 \sqrt{lim_{x \to 1} x} + lim_{x \to 1} 15 x^{\frac{3}{2}}) \cdot lim_{x \to 1} \sqrt{x} - lim_{x \to 1} 1}{lim_{x \to 1} \sqrt{x}}}{lim_{x \to 1} \frac{3 \sqrt{x} - 1}{\sqrt{x}}}$

Step 8.9

Move the term $15$ outside of the limit because it is constant with respect to $x$ .

$2 \frac{\frac{1}{2} \cdot \frac{(- 1 \cdot 3 \cdot 2 \sqrt{lim_{x \to 1} x} + 15 lim_{x \to 1} x^{\frac{3}{2}}) \cdot lim_{x \to 1} \sqrt{x} - lim_{x \to 1} 1}{lim_{x \to 1} \sqrt{x}}}{lim_{x \to 1} \frac{3 \sqrt{x} - 1}{\sqrt{x}}}$

Step 8.10

Move the exponent $\frac{3}{2}$ from $x^{\frac{3}{2}}$ outside the limit using the Limits Power Rule.

$2 \frac{\frac{1}{2} \cdot \frac{(- 1 \cdot 3 \cdot 2 \sqrt{lim_{x \to 1} x} + 15 {(lim_{x \to 1} x)}^{\frac{3}{2}}) \cdot lim_{x \to 1} \sqrt{x} - lim_{x \to 1} 1}{lim_{x \to 1} \sqrt{x}}}{lim_{x \to 1} \frac{3 \sqrt{x} - 1}{\sqrt{x}}}$

Step 8.11

Move the limit under the radical sign.

$2 \frac{\frac{1}{2} \cdot \frac{(- 1 \cdot 3 \cdot 2 \sqrt{lim_{x \to 1} x} + 15 {(lim_{x \to 1} x)}^{\frac{3}{2}}) \cdot \sqrt{lim_{x \to 1} x} - lim_{x \to 1} 1}{lim_{x \to 1} \sqrt{x}}}{lim_{x \to 1} \frac{3 \sqrt{x} - 1}{\sqrt{x}}}$

Step 8.12

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$2 \frac{\frac{1}{2} \cdot \frac{(- 1 \cdot 3 \cdot 2 \sqrt{lim_{x \to 1} x} + 15 {(lim_{x \to 1} x)}^{\frac{3}{2}}) \cdot \sqrt{lim_{x \to 1} x} - 1 \cdot 1}{lim_{x \to 1} \sqrt{x}}}{lim_{x \to 1} \frac{3 \sqrt{x} - 1}{\sqrt{x}}}$

Step 8.13

Move the limit under the radical sign.

$2 \frac{\frac{1}{2} \cdot \frac{(- 1 \cdot 3 \cdot 2 \sqrt{lim_{x \to 1} x} + 15 {(lim_{x \to 1} x)}^{\frac{3}{2}}) \cdot \sqrt{lim_{x \to 1} x} - 1 \cdot 1}{\sqrt{lim_{x \to 1} x}}}{lim_{x \to 1} \frac{3 \sqrt{x} - 1}{\sqrt{x}}}$

Step 8.14

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

Step 8.15

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

Step 8.16

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

Step 8.17

Move the limit under the radical sign.

Step 8.18

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

Step 8.19

Move the limit under the radical sign.

Step 9

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$2 \frac{\frac{1}{2} \cdot \frac{(- 1 \cdot 3 \cdot 2 \sqrt{1} + 15 {(lim_{x \to 1} x)}^{\frac{3}{2}}) \cdot \sqrt{lim_{x \to 1} x} - 1 \cdot 1}{\sqrt{lim_{x \to 1} x}}}{\frac{3 \sqrt{lim_{x \to 1} x} - 1 \cdot 1}{\sqrt{lim_{x \to 1} x}}}$

Step 9.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$2 \frac{\frac{1}{2} \cdot \frac{(- 1 \cdot 3 \cdot 2 \sqrt{1} + 15 \cdot 1^{\frac{3}{2}}) \cdot \sqrt{lim_{x \to 1} x} - 1 \cdot 1}{\sqrt{lim_{x \to 1} x}}}{\frac{3 \sqrt{lim_{x \to 1} x} - 1 \cdot 1}{\sqrt{lim_{x \to 1} x}}}$

Step 9.3

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$2 \frac{\frac{1}{2} \cdot \frac{(- 1 \cdot 3 \cdot 2 \sqrt{1} + 15 \cdot 1^{\frac{3}{2}}) \cdot \sqrt{1} - 1 \cdot 1}{\sqrt{lim_{x \to 1} x}}}{\frac{3 \sqrt{lim_{x \to 1} x} - 1 \cdot 1}{\sqrt{lim_{x \to 1} x}}}$

Step 9.4

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$2 \frac{\frac{1}{2} \cdot \frac{(- 1 \cdot 3 \cdot 2 \sqrt{1} + 15 \cdot 1^{\frac{3}{2}}) \cdot \sqrt{1} - 1 \cdot 1}{\sqrt{1}}}{\frac{3 \sqrt{lim_{x \to 1} x} - 1 \cdot 1}{\sqrt{lim_{x \to 1} x}}}$

Step 9.5

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$2 \frac{\frac{1}{2} \cdot \frac{(- 1 \cdot 3 \cdot 2 \sqrt{1} + 15 \cdot 1^{\frac{3}{2}}) \cdot \sqrt{1} - 1 \cdot 1}{\sqrt{1}}}{\frac{3 \sqrt{1} - 1 \cdot 1}{\sqrt{lim_{x \to 1} x}}}$

Step 9.6

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$2 \frac{\frac{1}{2} \cdot \frac{(- 1 \cdot 3 \cdot 2 \sqrt{1} + 15 \cdot 1^{\frac{3}{2}}) \cdot \sqrt{1} - 1 \cdot 1}{\sqrt{1}}}{\frac{3 \sqrt{1} - 1 \cdot 1}{\sqrt{1}}}$

Step 10

Simplify the answer.

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Step 10.1

Multiply the numerator by the reciprocal of the denominator.

$2 (\frac{1}{2} \cdot \frac{(- 1 \cdot 3 \cdot 2 \sqrt{1} + 15 \cdot 1^{\frac{3}{2}}) \cdot \sqrt{1} - 1 \cdot 1}{\sqrt{1}} \frac{\sqrt{1}}{3 \sqrt{1} - 1 \cdot 1})$

Step 10.2

Simplify the numerator.

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Step 10.2.1

Simplify each term.

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Step 10.2.1.1

Multiply $- 1 \cdot 3 \cdot 2$ .

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Step 10.2.1.1.1

Multiply $- 1$ by $3$ .

$2 (\frac{1}{2} \cdot \frac{(- 3 \cdot 2 \sqrt{1} + 15 \cdot 1^{\frac{3}{2}}) \cdot \sqrt{1} - 1 \cdot 1}{\sqrt{1}} \frac{\sqrt{1}}{3 \sqrt{1} - 1 \cdot 1})$

Step 10.2.1.1.2

Multiply $- 3$ by $2$ .

$2 (\frac{1}{2} \cdot \frac{(- 6 \sqrt{1} + 15 \cdot 1^{\frac{3}{2}}) \cdot \sqrt{1} - 1 \cdot 1}{\sqrt{1}} \frac{\sqrt{1}}{3 \sqrt{1} - 1 \cdot 1})$

Step 10.2.1.2

Any root of $1$ is $1$ .

$2 (\frac{1}{2} \cdot \frac{(- 6 \cdot 1 + 15 \cdot 1^{\frac{3}{2}}) \cdot \sqrt{1} - 1 \cdot 1}{\sqrt{1}} \frac{\sqrt{1}}{3 \sqrt{1} - 1 \cdot 1})$

Step 10.2.1.3

Multiply $- 6$ by $1$ .

$2 (\frac{1}{2} \cdot \frac{(- 6 + 15 \cdot 1^{\frac{3}{2}}) \cdot \sqrt{1} - 1 \cdot 1}{\sqrt{1}} \frac{\sqrt{1}}{3 \sqrt{1} - 1 \cdot 1})$

Step 10.2.1.4

One to any power is one.

$2 (\frac{1}{2} \cdot \frac{(- 6 + 15 \cdot 1) \cdot \sqrt{1} - 1 \cdot 1}{\sqrt{1}} \frac{\sqrt{1}}{3 \sqrt{1} - 1 \cdot 1})$

Step 10.2.1.5

Multiply $15$ by $1$ .

$2 (\frac{1}{2} \cdot \frac{(- 6 + 15) \cdot \sqrt{1} - 1 \cdot 1}{\sqrt{1}} \frac{\sqrt{1}}{3 \sqrt{1} - 1 \cdot 1})$

Step 10.2.2

Add $- 6$ and $15$ .

$2 (\frac{1}{2} \cdot \frac{9 \cdot \sqrt{1} - 1 \cdot 1}{\sqrt{1}} \frac{\sqrt{1}}{3 \sqrt{1} - 1 \cdot 1})$

Step 10.2.3

Any root of $1$ is $1$ .

$2 (\frac{1}{2} \cdot \frac{9 \cdot 1 - 1 \cdot 1}{\sqrt{1}} \frac{\sqrt{1}}{3 \sqrt{1} - 1 \cdot 1})$

Step 10.2.4

Multiply $9$ by $1$ .

$2 (\frac{1}{2} \cdot \frac{9 - 1 \cdot 1}{\sqrt{1}} \frac{\sqrt{1}}{3 \sqrt{1} - 1 \cdot 1})$

Step 10.2.5

Multiply $- 1$ by $1$ .

$2 (\frac{1}{2} \cdot \frac{9 - 1}{\sqrt{1}} \frac{\sqrt{1}}{3 \sqrt{1} - 1 \cdot 1})$

Step 10.2.6

Subtract $1$ from $9$ .

$2 (\frac{1}{2} \cdot \frac{8}{\sqrt{1}} \frac{\sqrt{1}}{3 \sqrt{1} - 1 \cdot 1})$

Step 10.3

Any root of $1$ is $1$ .

$2 (\frac{1}{2} \cdot \frac{8}{1} \frac{\sqrt{1}}{3 \sqrt{1} - 1 \cdot 1})$

Step 10.4

Cancel the common factor of $2$ .

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Step 10.4.1

Factor $2$ out of $8$ .

$2 (\frac{1}{2} \cdot \frac{2 (4)}{1} \frac{\sqrt{1}}{3 \sqrt{1} - 1 \cdot 1})$

Step 10.4.2

Cancel the common factor.

$2 (\frac{1}{2} \cdot \frac{2 \cdot 4}{1} \frac{\sqrt{1}}{3 \sqrt{1} - 1 \cdot 1})$

Step 10.4.3

Rewrite the expression.

$2 (4 \frac{\sqrt{1}}{3 \sqrt{1} - 1 \cdot 1})$

Step 10.5

Any root of $1$ is $1$ .

$2 (4 \frac{1}{3 \sqrt{1} - 1 \cdot 1})$

Step 10.6

Simplify the denominator.

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Step 10.6.1

Any root of $1$ is $1$ .

$2 (4 \frac{1}{3 \cdot 1 - 1 \cdot 1})$

Step 10.6.2

Multiply $3$ by $1$ .

$2 (4 \frac{1}{3 - 1 \cdot 1})$

Step 10.6.3

Multiply $- 1$ by $1$ .

$2 (4 \frac{1}{3 - 1})$

Step 10.6.4

Subtract $1$ from $3$ .

$2 (4 (\frac{1}{2}))$

Step 10.7

Cancel the common factor of $2$ .

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Step 10.7.1

Factor $2$ out of $4$ .

$2 (2 (2) \frac{1}{2})$

Step 10.7.2

Cancel the common factor.

$2 (2 \cdot 2 \frac{1}{2})$

Step 10.7.3

Rewrite the expression.

$2 \cdot 2$

Step 10.8

Multiply $2$ by $2$ .

$4$