Calculus Examples

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

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.

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

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 $1$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 1.2.6

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

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Step 1.2.6.1

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

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

Step 1.2.6.2

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

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

Step 1.2.7

Simplify the answer.

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Step 1.2.7.1

Simplify each term.

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Step 1.2.7.1.1

One to any power is one.

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

Step 1.2.7.1.2

One to any power is one.

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

Step 1.2.7.1.3

Multiply $- 2$ by $1$ .

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

Step 1.2.7.2

Subtract $2$ from $1$ .

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

Step 1.2.7.3

Add $- 1$ and $1$ .

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

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

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

Step 1.3.1.2

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

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

Step 1.3.1.3

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

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

Step 1.3.2

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

$\frac{0}{1^{3} - 1 \cdot 1}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

One to any power is one.

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

Step 1.3.3.1.2

Multiply $- 1$ by $1$ .

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

Step 1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.3.4

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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

Multiply $2$ by $- 2$ .

$lim_{x \to 1} \frac{3 x^{2} - 4 x + \frac{d}{d x} [1]}{\frac{d}{d x} [x^{3} - 1]}$

Step 3.5

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

$lim_{x \to 1} \frac{3 x^{2} - 4 x + 0}{\frac{d}{d x} [x^{3} - 1]}$

Step 3.6

Add $3 x^{2} - 4 x$ and $0$ .

$lim_{x \to 1} \frac{3 x^{2} - 4 x}{\frac{d}{d x} [x^{3} - 1]}$

Step 3.7

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

$lim_{x \to 1} \frac{3 x^{2} - 4 x}{\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 1]}$

Step 3.8

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

$lim_{x \to 1} \frac{3 x^{2} - 4 x}{3 x^{2} + \frac{d}{d x} [- 1]}$

Step 3.9

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

$lim_{x \to 1} \frac{3 x^{2} - 4 x}{3 x^{2} + 0}$

Step 3.10

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

$lim_{x \to 1} \frac{3 x^{2} - 4 x}{3 x^{2}}$

Step 4

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} lim_{x \to 1} \frac{3 x^{2} - 4 x}{x^{2}}$

Step 5

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

$\frac{1}{3} \cdot \frac{lim_{x \to 1} 3 x^{2} - 4 x}{lim_{x \to 1} x^{2}}$

Step 6

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

$\frac{1}{3} \cdot \frac{lim_{x \to 1} 3 x^{2} - lim_{x \to 1} 4 x}{lim_{x \to 1} x^{2}}$

Step 7

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

$\frac{1}{3} \cdot \frac{3 lim_{x \to 1} x^{2} - lim_{x \to 1} 4 x}{lim_{x \to 1} x^{2}}$

Step 8

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

$\frac{1}{3} \cdot \frac{3 {(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 4 x}{lim_{x \to 1} x^{2}}$

Step 9

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} \cdot \frac{3 {(lim_{x \to 1} x)}^{2} - 4 lim_{x \to 1} x}{lim_{x \to 1} x^{2}}$

Step 10

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

$\frac{1}{3} \cdot \frac{3 {(lim_{x \to 1} x)}^{2} - 4 lim_{x \to 1} x}{{(lim_{x \to 1} x)}^{2}}$

Step 11

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

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Step 11.1

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

$\frac{1}{3} \cdot \frac{3 \cdot 1^{2} - 4 lim_{x \to 1} x}{{(lim_{x \to 1} x)}^{2}}$

Step 11.2

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

$\frac{1}{3} \cdot \frac{3 \cdot 1^{2} - 4 \cdot 1}{{(lim_{x \to 1} x)}^{2}}$

Step 11.3

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

$\frac{1}{3} \cdot \frac{3 \cdot 1^{2} - 4 \cdot 1}{1^{2}}$

Step 12

Simplify the answer.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

One to any power is one.

$\frac{1}{3} \cdot \frac{3 \cdot 1 - 4 \cdot 1}{1^{2}}$

Step 12.1.2

Multiply $3$ by $1$ .

$\frac{1}{3} \cdot \frac{3 - 4 \cdot 1}{1^{2}}$

Step 12.1.3

Multiply $- 4$ by $1$ .

$\frac{1}{3} \cdot \frac{3 - 4}{1^{2}}$

Step 12.1.4

Subtract $4$ from $3$ .

$\frac{1}{3} \cdot \frac{- 1}{1^{2}}$

Step 12.2

One to any power is one.

$\frac{1}{3} \cdot \frac{- 1}{1}$

Step 12.3

Divide $- 1$ by $1$ .

$\frac{1}{3} \cdot - 1$

Step 12.4

Combine $\frac{1}{3}$ and $- 1$ .

$\frac{- 1}{3}$

Step 12.5

Move the negative in front of the fraction.

$- \frac{1}{3}$