Calculus Examples

$lim_{x \to 0} \frac{\cos (x) - 1 + \frac{x^{2}}{2}}{8 x^{4}}$

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.

$\frac{lim_{x \to 0} \cos (x) - 1 + \frac{x^{2}}{2}}{lim_{x \to 0} 8 x^{4}}$

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 $0$ .

$\frac{lim_{x \to 0} \cos (x) - lim_{x \to 0} 1 + lim_{x \to 0} \frac{x^{2}}{2}}{lim_{x \to 0} 8 x^{4}}$

Step 1.2.2

Move the limit inside the trig function because cosine is continuous.

$\frac{\cos (lim_{x \to 0} x) - lim_{x \to 0} 1 + lim_{x \to 0} \frac{x^{2}}{2}}{lim_{x \to 0} 8 x^{4}}$

Step 1.2.3

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

$\frac{\cos (lim_{x \to 0} x) - 1 \cdot 1 + lim_{x \to 0} \frac{x^{2}}{2}}{lim_{x \to 0} 8 x^{4}}$

Step 1.2.4

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

$\frac{\cos (lim_{x \to 0} x) - 1 \cdot 1 + \frac{1}{2} lim_{x \to 0} x^{2}}{lim_{x \to 0} 8 x^{4}}$

Step 1.2.5

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

$\frac{\cos (lim_{x \to 0} x) - 1 \cdot 1 + \frac{1}{2} {(lim_{x \to 0} x)}^{2}}{lim_{x \to 0} 8 x^{4}}$

Step 1.2.6

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

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Step 1.2.6.1

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

$\frac{\cos (0) - 1 \cdot 1 + \frac{1}{2} {(lim_{x \to 0} x)}^{2}}{lim_{x \to 0} 8 x^{4}}$

Step 1.2.6.2

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

$\frac{\cos (0) - 1 \cdot 1 + \frac{1}{2} \cdot 0^{2}}{lim_{x \to 0} 8 x^{4}}$

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

The exact value of $\cos (0)$ is $1$ .

$\frac{1 - 1 \cdot 1 + \frac{1}{2} \cdot 0^{2}}{lim_{x \to 0} 8 x^{4}}$

Step 1.2.7.1.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1 + \frac{1}{2} \cdot 0^{2}}{lim_{x \to 0} 8 x^{4}}$

Step 1.2.7.1.3

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

$\frac{1 - 1 + \frac{1}{2} \cdot 0}{lim_{x \to 0} 8 x^{4}}$

Step 1.2.7.1.4

Multiply $\frac{1}{2}$ by $0$ .

$\frac{1 - 1 + 0}{lim_{x \to 0} 8 x^{4}}$

Step 1.2.7.2

Subtract $1$ from $1$ .

$\frac{0 + 0}{lim_{x \to 0} 8 x^{4}}$

Step 1.2.7.3

Add $0$ and $0$ .

$\frac{0}{lim_{x \to 0} 8 x^{4}}$

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

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

$\frac{0}{8 lim_{x \to 0} x^{4}}$

Step 1.3.1.2

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\frac{0}{8 {(lim_{x \to 0} x)}^{4}}$

Step 1.3.2

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

$\frac{0}{8 \cdot 0^{4}}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

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

$\frac{0}{8 \cdot 0}$

Step 1.3.3.2

Multiply $8$ by $0$ .

$\frac{0}{0}$

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.

$lim_{x \to 0} \frac{\cos (x) - 1 + \frac{x^{2}}{2}}{8 x^{4}} = lim_{x \to 0} \frac{\frac{d}{d x} [\cos (x) - 1 + \frac{x^{2}}{2}]}{\frac{d}{d x} [8 x^{4}]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\cos (x) - 1 + \frac{x^{2}}{2}]}{\frac{d}{d x} [8 x^{4}]}$

Step 3.2

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

$lim_{x \to 0} \frac{\frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- 1] + \frac{d}{d x} [\frac{x^{2}}{2}]}{\frac{d}{d x} [8 x^{4}]}$

Step 3.3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$lim_{x \to 0} \frac{- \sin (x) + \frac{d}{d x} [- 1] + \frac{d}{d x} [\frac{x^{2}}{2}]}{\frac{d}{d x} [8 x^{4}]}$

Step 3.4

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

$lim_{x \to 0} \frac{- \sin (x) + 0 + \frac{d}{d x} [\frac{x^{2}}{2}]}{\frac{d}{d x} [8 x^{4}]}$

Step 3.5

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

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Step 3.5.1

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

$lim_{x \to 0} \frac{- \sin (x) + 0 + \frac{1}{2} \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [8 x^{4}]}$

Step 3.5.2

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

$lim_{x \to 0} \frac{- \sin (x) + 0 + \frac{1}{2} (2 x)}{\frac{d}{d x} [8 x^{4}]}$

Step 3.5.3

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

$lim_{x \to 0} \frac{- \sin (x) + 0 + \frac{2}{2} x}{\frac{d}{d x} [8 x^{4}]}$

Step 3.5.4

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

$lim_{x \to 0} \frac{- \sin (x) + 0 + \frac{2 x}{2}}{\frac{d}{d x} [8 x^{4}]}$

Step 3.5.5

Cancel the common factor of $2$ .

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Step 3.5.5.1

Cancel the common factor.

$lim_{x \to 0} \frac{- \sin (x) + 0 + \frac{2 x}{2}}{\frac{d}{d x} [8 x^{4}]}$

Step 3.5.5.2

Divide $x$ by $1$ .

$lim_{x \to 0} \frac{- \sin (x) + 0 + x}{\frac{d}{d x} [8 x^{4}]}$

Step 3.6

Simplify.

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Step 3.6.1

Add $- \sin (x)$ and $0$ .

$lim_{x \to 0} \frac{- \sin (x) + x}{\frac{d}{d x} [8 x^{4}]}$

Step 3.6.2

Reorder terms.

$lim_{x \to 0} \frac{x - \sin (x)}{\frac{d}{d x} [8 x^{4}]}$

Step 3.7

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{4}$ with respect to $x$ is $8 \frac{d}{d x} [x^{4}]$ .

$lim_{x \to 0} \frac{x - \sin (x)}{8 \frac{d}{d x} [x^{4}]}$

Step 3.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$lim_{x \to 0} \frac{x - \sin (x)}{8 (4 x^{3})}$

Step 3.9

Multiply $4$ by $8$ .

$lim_{x \to 0} \frac{x - \sin (x)}{32 x^{3}}$

Step 4

Move the term $\frac{1}{32}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{32} lim_{x \to 0} \frac{x - \sin (x)}{x^{3}}$

Step 5

Apply L'Hospital's rule.

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Step 5.1

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

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Step 5.1.1

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

$\frac{1}{32} \cdot \frac{lim_{x \to 0} x - \sin (x)}{lim_{x \to 0} x^{3}}$

Step 5.1.2

Evaluate the limit of the numerator.

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Step 5.1.2.1

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

$\frac{1}{32} \cdot \frac{lim_{x \to 0} x - lim_{x \to 0} \sin (x)}{lim_{x \to 0} x^{3}}$

Step 5.1.2.2

Move the limit inside the trig function because sine is continuous.

$\frac{1}{32} \cdot \frac{lim_{x \to 0} x - \sin (lim_{x \to 0} x)}{lim_{x \to 0} x^{3}}$

Step 5.1.2.3

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

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Step 5.1.2.3.1

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

$\frac{1}{32} \cdot \frac{0 - \sin (lim_{x \to 0} x)}{lim_{x \to 0} x^{3}}$

Step 5.1.2.3.2

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

$\frac{1}{32} \cdot \frac{0 - \sin (0)}{lim_{x \to 0} x^{3}}$

Step 5.1.2.4

Simplify the answer.

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Step 5.1.2.4.1

Simplify each term.

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Step 5.1.2.4.1.1

The exact value of $\sin (0)$ is $0$ .

$\frac{1}{32} \cdot \frac{0 - 0}{lim_{x \to 0} x^{3}}$

Step 5.1.2.4.1.2

Multiply $- 1$ by $0$ .

$\frac{1}{32} \cdot \frac{0 + 0}{lim_{x \to 0} x^{3}}$

Step 5.1.2.4.2

Add $0$ and $0$ .

$\frac{1}{32} \cdot \frac{0}{lim_{x \to 0} x^{3}}$

Step 5.1.3

Evaluate the limit of the denominator.

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Step 5.1.3.1

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

$\frac{1}{32} \cdot \frac{0}{{(lim_{x \to 0} x)}^{3}}$

Step 5.1.3.2

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

$\frac{1}{32} \cdot \frac{0}{0^{3}}$

Step 5.1.3.3

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

$\frac{1}{32} \cdot \frac{0}{0}$

Step 5.1.3.4

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

Undefined

$\frac{1}{32} \cdot \frac{0}{0}$

Step 5.1.4

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

Undefined

$\frac{1}{32} \cdot \frac{0}{0}$

Step 5.2

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.

$lim_{x \to 0} \frac{x - \sin (x)}{x^{3}} = lim_{x \to 0} \frac{\frac{d}{d x} [x - \sin (x)]}{\frac{d}{d x} [x^{3}]}$

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

$\frac{1}{32} lim_{x \to 0} \frac{\frac{d}{d x} [x - \sin (x)]}{\frac{d}{d x} [x^{3}]}$

Step 5.3.2

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

$\frac{1}{32} lim_{x \to 0} \frac{\frac{d}{d x} [x] + \frac{d}{d x} [- \sin (x)]}{\frac{d}{d x} [x^{3}]}$

Step 5.3.3

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

$\frac{1}{32} lim_{x \to 0} \frac{1 + \frac{d}{d x} [- \sin (x)]}{\frac{d}{d x} [x^{3}]}$

Step 5.3.4

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

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Step 5.3.4.1

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

$\frac{1}{32} lim_{x \to 0} \frac{1 - \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x^{3}]}$

Step 5.3.4.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\frac{1}{32} lim_{x \to 0} \frac{1 - \cos (x)}{\frac{d}{d x} [x^{3}]}$

Step 5.3.5

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

$\frac{1}{32} lim_{x \to 0} \frac{1 - \cos (x)}{3 x^{2}}$

Step 6

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

$\frac{1}{32} \cdot \frac{1}{3} lim_{x \to 0} \frac{1 - \cos (x)}{x^{2}}$

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.

$\frac{1}{32} \cdot \frac{1}{3} \frac{lim_{x \to 0} 1 - \cos (x)}{lim_{x \to 0} x^{2}}$

Step 7.1.2

Evaluate the limit of the numerator.

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Step 7.1.2.1

Evaluate the limit.

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Step 7.1.2.1.1

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

$\frac{1}{32} \cdot \frac{1}{3} \frac{lim_{x \to 0} 1 - lim_{x \to 0} \cos (x)}{lim_{x \to 0} x^{2}}$

Step 7.1.2.1.2

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

$\frac{1}{32} \cdot \frac{1}{3} \frac{1 - lim_{x \to 0} \cos (x)}{lim_{x \to 0} x^{2}}$

Step 7.1.2.1.3

Move the limit inside the trig function because cosine is continuous.

$\frac{1}{32} \cdot \frac{1}{3} \frac{1 - \cos (lim_{x \to 0} x)}{lim_{x \to 0} x^{2}}$

Step 7.1.2.2

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

$\frac{1}{32} \cdot \frac{1}{3} \frac{1 - \cos (0)}{lim_{x \to 0} x^{2}}$

Step 7.1.2.3

Simplify the answer.

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Step 7.1.2.3.1

Simplify each term.

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Step 7.1.2.3.1.1

The exact value of $\cos (0)$ is $1$ .

$\frac{1}{32} \cdot \frac{1}{3} \frac{1 - 1 \cdot 1}{lim_{x \to 0} x^{2}}$

Step 7.1.2.3.1.2

Multiply $- 1$ by $1$ .

$\frac{1}{32} \cdot \frac{1}{3} \frac{1 - 1}{lim_{x \to 0} x^{2}}$

Step 7.1.2.3.2

Subtract $1$ from $1$ .

$\frac{1}{32} \cdot \frac{1}{3} \frac{0}{lim_{x \to 0} x^{2}}$

Step 7.1.3

Evaluate the limit of the denominator.

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Step 7.1.3.1

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

$\frac{1}{32} \cdot \frac{1}{3} \frac{0}{{(lim_{x \to 0} x)}^{2}}$

Step 7.1.3.2

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

$\frac{1}{32} \cdot \frac{1}{3} \frac{0}{0^{2}}$

Step 7.1.3.3

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

$\frac{1}{32} \cdot \frac{1}{3} \frac{0}{0}$

Step 7.1.3.4

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

Undefined

$\frac{1}{32} \cdot \frac{1}{3} \frac{0}{0}$

Step 7.1.4

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

Undefined

$\frac{1}{32} \cdot \frac{1}{3} \frac{0}{0}$

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.

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

Step 7.3

Find the derivative of the numerator and denominator.

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Step 7.3.1

Differentiate the numerator and denominator.

$\frac{1}{32} \cdot \frac{1}{3} lim_{x \to 0} \frac{\frac{d}{d x} [1 - \cos (x)]}{\frac{d}{d x} [x^{2}]}$

Step 7.3.2

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

$\frac{1}{32} \cdot \frac{1}{3} lim_{x \to 0} \frac{\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (x)]}{\frac{d}{d x} [x^{2}]}$

Step 7.3.3

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

$\frac{1}{32} \cdot \frac{1}{3} lim_{x \to 0} \frac{0 + \frac{d}{d x} [- \cos (x)]}{\frac{d}{d x} [x^{2}]}$

Step 7.3.4

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

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Step 7.3.4.1

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

$\frac{1}{32} \cdot \frac{1}{3} lim_{x \to 0} \frac{0 - \frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [x^{2}]}$

Step 7.3.4.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$\frac{1}{32} \cdot \frac{1}{3} lim_{x \to 0} \frac{0 - - 1 \sin (x)}{\frac{d}{d x} [x^{2}]}$

Step 7.3.4.3

Multiply $- 1$ by $- 1$ .

$\frac{1}{32} \cdot \frac{1}{3} lim_{x \to 0} \frac{0 + 1 \sin (x)}{\frac{d}{d x} [x^{2}]}$

Step 7.3.4.4

Multiply $\sin (x)$ by $1$ .

$\frac{1}{32} \cdot \frac{1}{3} lim_{x \to 0} \frac{0 + \sin (x)}{\frac{d}{d x} [x^{2}]}$

Step 7.3.5

Add $0$ and $\sin (x)$ .

$\frac{1}{32} \cdot \frac{1}{3} lim_{x \to 0} \frac{\sin (x)}{\frac{d}{d x} [x^{2}]}$

Step 7.3.6

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

$\frac{1}{32} \cdot \frac{1}{3} lim_{x \to 0} \frac{\sin (x)}{2 x}$

Step 8

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

$\frac{1}{32} \cdot \frac{1}{3} \frac{1}{2} lim_{x \to 0} \frac{\sin (x)}{x}$

Step 9

Since $\cos (x) \leq \frac{\sin (x)}{x} \leq 1$ and $lim_{x \to 0} \cos (x) = lim_{x \to 0} 1 = 1$ , apply the squeeze theorem.

$\frac{1}{32} \cdot \frac{1}{3} \frac{1}{2} \cdot 1$

Step 10

Simplify the answer.

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Step 10.1

Multiply $\frac{1}{32} \cdot \frac{1}{3}$ .

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Step 10.1.1

Multiply $\frac{1}{32}$ by $\frac{1}{3}$ .

$\frac{1}{32 \cdot 3} \cdot \frac{1}{2} \cdot 1$

Step 10.1.2

Multiply $32$ by $3$ .

$\frac{1}{96} \cdot \frac{1}{2} \cdot 1$

Step 10.2

Multiply $\frac{1}{96} \cdot \frac{1}{2}$ .

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Step 10.2.1

Multiply $\frac{1}{96}$ by $\frac{1}{2}$ .

$\frac{1}{96 \cdot 2} \cdot 1$

Step 10.2.2

Multiply $96$ by $2$ .

$\frac{1}{192} \cdot 1$

Step 10.3

Multiply $\frac{1}{192}$ by $1$ .

$\frac{1}{192}$