Calculus Examples

$lim_{x \to 0} \frac{8}{x} - \frac{8}{x^{2} + x}$

Step 1

Combine terms.

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Step 1.1

To write $\frac{8}{x}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + x}{x^{2} + x}$ .

$lim_{x \to 0} \frac{8}{x} \cdot \frac{x^{2} + x}{x^{2} + x} - \frac{8}{x^{2} + x}$

Step 1.2

To write $- \frac{8}{x^{2} + x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$lim_{x \to 0} \frac{8}{x} \cdot \frac{x^{2} + x}{x^{2} + x} - \frac{8}{x^{2} + x} \cdot \frac{x}{x}$

Step 1.3

Write each expression with a common denominator of $x (x^{2} + x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{8}{x}$ by $\frac{x^{2} + x}{x^{2} + x}$ .

$lim_{x \to 0} \frac{8 (x^{2} + x)}{x (x^{2} + x)} - \frac{8}{x^{2} + x} \cdot \frac{x}{x}$

Step 1.3.2

Multiply $\frac{8}{x^{2} + x}$ by $\frac{x}{x}$ .

$lim_{x \to 0} \frac{8 (x^{2} + x)}{x (x^{2} + x)} - \frac{8 x}{(x^{2} + x) x}$

Step 1.3.3

Reorder the factors of $(x^{2} + x) x$ .

$lim_{x \to 0} \frac{8 (x^{2} + x)}{x (x^{2} + x)} - \frac{8 x}{x (x^{2} + x)}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{8 (x^{2} + x) - 8 x}{x (x^{2} + x)}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$\frac{lim_{x \to 0} 8 (x^{2} + x) - 8 x}{lim_{x \to 0} x (x^{2} + x)}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} 8 (x^{2} + x) - lim_{x \to 0} 8 x}{lim_{x \to 0} x (x^{2} + x)}$

Step 2.1.2.2

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{8 lim_{x \to 0} x^{2} + x - lim_{x \to 0} 8 x}{lim_{x \to 0} x (x^{2} + x)}$

Step 2.1.2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{8 (lim_{x \to 0} x^{2} + lim_{x \to 0} x) - lim_{x \to 0} 8 x}{lim_{x \to 0} x (x^{2} + x)}$

Step 2.1.2.4

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

$\frac{8 ({(lim_{x \to 0} x)}^{2} + lim_{x \to 0} x) - lim_{x \to 0} 8 x}{lim_{x \to 0} x (x^{2} + x)}$

Step 2.1.2.5

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{8 ({(lim_{x \to 0} x)}^{2} + lim_{x \to 0} x) - 8 lim_{x \to 0} x}{lim_{x \to 0} x (x^{2} + x)}$

Step 2.1.2.6

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 2.1.2.6.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{8 (0^{2} + lim_{x \to 0} x) - 8 lim_{x \to 0} x}{lim_{x \to 0} x (x^{2} + x)}$

Step 2.1.2.6.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{8 (0^{2} + 0) - 8 lim_{x \to 0} x}{lim_{x \to 0} x (x^{2} + x)}$

Step 2.1.2.6.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{8 (0^{2} + 0) - 8 \cdot 0}{lim_{x \to 0} x (x^{2} + x)}$

Step 2.1.2.7

Simplify the answer.

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Step 2.1.2.7.1

Simplify each term.

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Step 2.1.2.7.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{8 (0 + 0) - 8 \cdot 0}{lim_{x \to 0} x (x^{2} + x)}$

Step 2.1.2.7.1.2

Add $0$ and $0$ .

$\frac{8 \cdot 0 - 8 \cdot 0}{lim_{x \to 0} x (x^{2} + x)}$

Step 2.1.2.7.1.3

Multiply $8$ by $0$ .

$\frac{0 - 8 \cdot 0}{lim_{x \to 0} x (x^{2} + x)}$

Step 2.1.2.7.1.4

Multiply $- 8$ by $0$ .

$\frac{0 + 0}{lim_{x \to 0} x (x^{2} + x)}$

Step 2.1.2.7.2

Add $0$ and $0$ .

$\frac{0}{lim_{x \to 0} x (x^{2} + x)}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

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

Step 2.1.3.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{0}{lim_{x \to 0} x \cdot (lim_{x \to 0} x^{2} + lim_{x \to 0} x)}$

Step 2.1.3.3

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

$\frac{0}{lim_{x \to 0} x \cdot ({(lim_{x \to 0} x)}^{2} + lim_{x \to 0} x)}$

Step 2.1.3.4

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 2.1.3.4.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{0 \cdot ({(lim_{x \to 0} x)}^{2} + lim_{x \to 0} x)}$

Step 2.1.3.4.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{0 \cdot (0^{2} + lim_{x \to 0} x)}$

Step 2.1.3.4.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{0 \cdot (0^{2} + 0)}$

Step 2.1.3.5

Simplify the answer.

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Step 2.1.3.5.1

Raising $0$ to any positive power yields $0$ .

$\frac{0}{0 \cdot (0 + 0)}$

Step 2.1.3.5.2

Add $0$ and $0$ .

$\frac{0}{0 \cdot 0}$

Step 2.1.3.5.3

Multiply $0$ by $0$ .

$\frac{0}{0}$

Step 2.1.3.5.4

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

Undefined

$\frac{0}{0}$

Step 2.1.3.6

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

Undefined

$\frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{0}{0}$

Step 2.2

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.

$lim_{x \to 0} \frac{8 (x^{2} + x) - 8 x}{x (x^{2} + x)} = lim_{x \to 0} \frac{\frac{d}{d x} [8 (x^{2} + x) - 8 x]}{\frac{d}{d x} [x (x^{2} + x)]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [8 (x^{2} + x) - 8 x]}{\frac{d}{d x} [x (x^{2} + x)]}$

Step 2.3.2

By the Sum Rule, the derivative of $8 (x^{2} + x) - 8 x$ with respect to $x$ is $\frac{d}{d x} [8 (x^{2} + x)] + \frac{d}{d x} [- 8 x]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [8 (x^{2} + x)] + \frac{d}{d x} [- 8 x]}{\frac{d}{d x} [x (x^{2} + x)]}$

Step 2.3.3

Evaluate $\frac{d}{d x} [8 (x^{2} + x)]$ .

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Step 2.3.3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 (x^{2} + x)$ with respect to $x$ is $8 \frac{d}{d x} [x^{2} + x]$ .

$lim_{x \to 0} \frac{8 \frac{d}{d x} [x^{2} + x] + \frac{d}{d x} [- 8 x]}{\frac{d}{d x} [x (x^{2} + x)]}$

Step 2.3.3.2

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

$lim_{x \to 0} \frac{8 (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [x]) + \frac{d}{d x} [- 8 x]}{\frac{d}{d x} [x (x^{2} + x)]}$

Step 2.3.3.3

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

$lim_{x \to 0} \frac{8 (2 x + \frac{d}{d x} [x]) + \frac{d}{d x} [- 8 x]}{\frac{d}{d x} [x (x^{2} + x)]}$

Step 2.3.3.4

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

$lim_{x \to 0} \frac{8 (2 x + 1) + \frac{d}{d x} [- 8 x]}{\frac{d}{d x} [x (x^{2} + x)]}$

Step 2.3.4

Evaluate $\frac{d}{d x} [- 8 x]$ .

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Step 2.3.4.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{8 (2 x + 1) - 8 \frac{d}{d x} [x]}{\frac{d}{d x} [x (x^{2} + x)]}$

Step 2.3.4.2

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

$lim_{x \to 0} \frac{8 (2 x + 1) - 8 \cdot 1}{\frac{d}{d x} [x (x^{2} + x)]}$

Step 2.3.4.3

Multiply $- 8$ by $1$ .

$lim_{x \to 0} \frac{8 (2 x + 1) - 8}{\frac{d}{d x} [x (x^{2} + x)]}$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Apply the distributive property.

$lim_{x \to 0} \frac{8 (2 x) + 8 \cdot 1 - 8}{\frac{d}{d x} [x (x^{2} + x)]}$

Step 2.3.5.2

Combine terms.

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Step 2.3.5.2.1

Multiply $2$ by $8$ .

$lim_{x \to 0} \frac{16 x + 8 \cdot 1 - 8}{\frac{d}{d x} [x (x^{2} + x)]}$

Step 2.3.5.2.2

Multiply $8$ by $1$ .

$lim_{x \to 0} \frac{16 x + 8 - 8}{\frac{d}{d x} [x (x^{2} + x)]}$

Step 2.3.5.2.3

Subtract $8$ from $8$ .

$lim_{x \to 0} \frac{16 x + 0}{\frac{d}{d x} [x (x^{2} + x)]}$

Step 2.3.5.2.4

Add $16 x$ and $0$ .

$lim_{x \to 0} \frac{16 x}{\frac{d}{d x} [x (x^{2} + x)]}$

Step 2.3.6

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 $f (x) = x$ and $g (x) = x^{2} + x$ .

$lim_{x \to 0} \frac{16 x}{x \frac{d}{d x} [x^{2} + x] + (x^{2} + x) \frac{d}{d x} [x]}$

Step 2.3.7

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

$lim_{x \to 0} \frac{16 x}{x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [x]) + (x^{2} + x) \frac{d}{d x} [x]}$

Step 2.3.8

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

$lim_{x \to 0} \frac{16 x}{x (2 x + \frac{d}{d x} [x]) + (x^{2} + x) \frac{d}{d x} [x]}$

Step 2.3.9

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

$lim_{x \to 0} \frac{16 x}{x (2 x + 1) + (x^{2} + x) \frac{d}{d x} [x]}$

Step 2.3.10

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

$lim_{x \to 0} \frac{16 x}{x (2 x + 1) + (x^{2} + x) \cdot 1}$

Step 2.3.11

Multiply $x^{2} + x$ by $1$ .

$lim_{x \to 0} \frac{16 x}{x (2 x + 1) + x^{2} + x}$

Step 2.3.12

Simplify.

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Step 2.3.12.1

Apply the distributive property.

$lim_{x \to 0} \frac{16 x}{x (2 x) + x \cdot 1 + x^{2} + x}$

Step 2.3.12.2

Combine terms.

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Step 2.3.12.2.1

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

$lim_{x \to 0} \frac{16 x}{2 (x^{1} x) + x \cdot 1 + x^{2} + x}$

Step 2.3.12.2.2

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

$lim_{x \to 0} \frac{16 x}{2 (x^{1} x^{1}) + x \cdot 1 + x^{2} + x}$

Step 2.3.12.2.3

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

$lim_{x \to 0} \frac{16 x}{2 x^{1 + 1} + x \cdot 1 + x^{2} + x}$

Step 2.3.12.2.4

Add $1$ and $1$ .

$lim_{x \to 0} \frac{16 x}{2 x^{2} + x \cdot 1 + x^{2} + x}$

Step 2.3.12.2.5

Multiply $x$ by $1$ .

$lim_{x \to 0} \frac{16 x}{2 x^{2} + x + x^{2} + x}$

Step 2.3.12.2.6

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

$lim_{x \to 0} \frac{16 x}{3 x^{2} + x + x}$

Step 2.3.12.2.7

Add $x$ and $x$ .

$lim_{x \to 0} \frac{16 x}{3 x^{2} + 2 x}$

Step 3

Move the term $16$ outside of the limit because it is constant with respect to $x$ .

$16 lim_{x \to 0} \frac{x}{3 x^{2} + 2 x}$

Step 4

Apply L'Hospital's rule.

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Step 4.1

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

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Step 4.1.1

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

$16 \frac{lim_{x \to 0} x}{lim_{x \to 0} 3 x^{2} + 2 x}$

Step 4.1.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$16 \frac{0}{lim_{x \to 0} 3 x^{2} + 2 x}$

Step 4.1.3

Evaluate the limit of the denominator.

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Step 4.1.3.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$16 \frac{0}{lim_{x \to 0} 3 x^{2} + lim_{x \to 0} 2 x}$

Step 4.1.3.2

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

$16 \frac{0}{3 lim_{x \to 0} x^{2} + lim_{x \to 0} 2 x}$

Step 4.1.3.3

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

$16 \frac{0}{3 {(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 2 x}$

Step 4.1.3.4

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

$16 \frac{0}{3 {(lim_{x \to 0} x)}^{2} + 2 lim_{x \to 0} x}$

Step 4.1.3.5

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 4.1.3.5.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$16 \frac{0}{3 \cdot 0^{2} + 2 lim_{x \to 0} x}$

Step 4.1.3.5.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$16 \frac{0}{3 \cdot 0^{2} + 2 \cdot 0}$

Step 4.1.3.6

Simplify the answer.

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Step 4.1.3.6.1

Simplify each term.

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Step 4.1.3.6.1.1

Raising $0$ to any positive power yields $0$ .

$16 \frac{0}{3 \cdot 0 + 2 \cdot 0}$

Step 4.1.3.6.1.2

Multiply $3$ by $0$ .

$16 \frac{0}{0 + 2 \cdot 0}$

Step 4.1.3.6.1.3

Multiply $2$ by $0$ .

$16 \frac{0}{0 + 0}$

Step 4.1.3.6.2

Add $0$ and $0$ .

$16 (\frac{0}{0})$

Step 4.1.3.6.3

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

Undefined

$16 (\frac{0}{0})$

Step 4.1.3.7

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

Undefined

$16 (\frac{0}{0})$

Step 4.1.4

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

Undefined

$16 (\frac{0}{0})$

Step 4.2

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.

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

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

$16 lim_{x \to 0} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [3 x^{2} + 2 x]}$

Step 4.3.2

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

$16 lim_{x \to 0} \frac{1}{\frac{d}{d x} [3 x^{2} + 2 x]}$

Step 4.3.3

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

$16 lim_{x \to 0} \frac{1}{\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [2 x]}$

Step 4.3.4

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 4.3.4.1

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

$16 lim_{x \to 0} \frac{1}{3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x]}$

Step 4.3.4.2

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

$16 lim_{x \to 0} \frac{1}{3 \cdot 2 x + \frac{d}{d x} [2 x]}$

Step 4.3.4.3

Multiply $2$ by $3$ .

$16 lim_{x \to 0} \frac{1}{6 x + \frac{d}{d x} [2 x]}$

Step 4.3.5

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

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Step 4.3.5.1

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

$16 lim_{x \to 0} \frac{1}{6 x + 2 \frac{d}{d x} [x]}$

Step 4.3.5.2

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

$16 lim_{x \to 0} \frac{1}{6 x + 2 \cdot 1}$

Step 4.3.5.3

Multiply $2$ by $1$ .

$16 lim_{x \to 0} \frac{1}{6 x + 2}$

Step 5

Evaluate the limit.

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Step 5.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$16 \frac{lim_{x \to 0} 1}{lim_{x \to 0} 6 x + 2}$

Step 5.2

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$16 \frac{1}{lim_{x \to 0} 6 x + 2}$

Step 5.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$16 \frac{1}{lim_{x \to 0} 6 x + lim_{x \to 0} 2}$

Step 5.4

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$16 \frac{1}{6 lim_{x \to 0} x + lim_{x \to 0} 2}$

Step 5.5

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$16 \frac{1}{6 lim_{x \to 0} x + 2}$

Step 6

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$16 \frac{1}{6 \cdot 0 + 2}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the denominator.

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Step 7.1.1

Multiply $6$ by $0$ .

$16 \frac{1}{0 + 2}$

Step 7.1.2

Add $0$ and $2$ .

$16 (\frac{1}{2})$

Step 7.2

Cancel the common factor of $2$ .

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Step 7.2.1

Factor $2$ out of $16$ .

$2 (8) \frac{1}{2}$

Step 7.2.2

Cancel the common factor.

$2 \cdot 8 \frac{1}{2}$

Step 7.2.3

Rewrite the expression.

$8$