Calculus Examples

$lim_{x \to 0} \frac{x^{2}}{\sin^{2} (4 x)}$

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.

$\frac{lim_{x \to 0} x^{2}}{lim_{x \to 0} \sin^{2} (4 x)}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

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

$\frac{{(lim_{x \to 0} x)}^{2}}{lim_{x \to 0} \sin^{2} (4 x)}$

Step 1.2.2

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

$\frac{0^{2}}{lim_{x \to 0} \sin^{2} (4 x)}$

Step 1.2.3

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

$\frac{0}{lim_{x \to 0} \sin^{2} (4 x)}$

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 exponent $2$ from $\sin^{2} (4 x)$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{x \to 0} \sin (4 x))}^{2}}$

Step 1.3.1.2

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

$\frac{0}{\sin^{2} (lim_{x \to 0} 4 x)}$

Step 1.3.1.3

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

$\frac{0}{\sin^{2} (4 lim_{x \to 0} x)}$

Step 1.3.2

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

$\frac{0}{\sin^{2} (4 \cdot 0)}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Multiply $4$ by $0$ .

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

Step 1.3.3.2

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

$\frac{0}{0^{2}}$

Step 1.3.3.3

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

$\frac{0}{0}$

Step 1.3.3.4

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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sin (4 x)$ .

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Step 3.3.1

To apply the Chain Rule, set $u_{1}$ as $\sin (4 x)$ .

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

Step 3.3.2

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

$lim_{x \to 0} \frac{2 x}{2 u_{1} \frac{d}{d x} [\sin (4 x)]}$

Step 3.3.3

Replace all occurrences of $u_{1}$ with $\sin (4 x)$ .

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

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 4 x$ .

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Step 3.4.1

To apply the Chain Rule, set $u_{2}$ as $4 x$ .

$lim_{x \to 0} \frac{2 x}{2 \sin (4 x) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [4 x])}$

Step 3.4.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$lim_{x \to 0} \frac{2 x}{2 \sin (4 x) (\cos (u_{2}) \frac{d}{d x} [4 x])}$

Step 3.4.3

Replace all occurrences of $u_{2}$ with $4 x$ .

$lim_{x \to 0} \frac{2 x}{2 \sin (4 x) (\cos (4 x) \frac{d}{d x} [4 x])}$

Step 3.5

Remove parentheses.

$lim_{x \to 0} \frac{2 x}{2 \sin (4 x) \cos (4 x) \frac{d}{d x} [4 x]}$

Step 3.6

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

$lim_{x \to 0} \frac{2 x}{2 \sin (4 x) \cos (4 x) (4 \frac{d}{d x} [x])}$

Step 3.7

Multiply $4$ by $2$ .

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

Step 3.8

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

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

Step 3.9

Multiply $8$ by $1$ .

$lim_{x \to 0} \frac{2 x}{8 \sin (4 x) \cos (4 x)}$

Step 3.10

Reorder the factors of $8 \sin (4 x) \cos (4 x)$ .

$lim_{x \to 0} \frac{2 x}{8 \cos (4 x) \sin (4 x)}$

Step 4

Evaluate the limit.

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Step 4.1

Cancel the common factor of $2$ and $8$ .

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Step 4.1.1

Factor $2$ out of $2 x$ .

$lim_{x \to 0} \frac{2 (x)}{8 \cos (4 x) \sin (4 x)}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $2$ out of $8 \cos (4 x) \sin (4 x)$ .

$lim_{x \to 0} \frac{2 (x)}{2 (4 \cos (4 x) \sin (4 x))}$

Step 4.1.2.2

Cancel the common factor.

$lim_{x \to 0} \frac{2 x}{2 (4 \cos (4 x) \sin (4 x))}$

Step 4.1.2.3

Rewrite the expression.

$lim_{x \to 0} \frac{x}{4 \cos (4 x) \sin (4 x)}$

Step 4.2

Move the term $\frac{1}{4}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{4} lim_{x \to 0} \frac{x}{\cos (4 x) \sin (4 x)}$

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.

$\frac{1}{4} \cdot \frac{lim_{x \to 0} x}{lim_{x \to 0} \cos (4 x) \sin (4 x)}$

Step 5.1.2

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

$\frac{1}{4} \cdot \frac{0}{lim_{x \to 0} \cos (4 x) \sin (4 x)}$

Step 5.1.3

Evaluate the limit of the denominator.

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Step 5.1.3.1

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

$\frac{1}{4} \cdot \frac{0}{lim_{x \to 0} \cos (4 x) \cdot lim_{x \to 0} \sin (4 x)}$

Step 5.1.3.2

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

$\frac{1}{4} \cdot \frac{0}{\cos (lim_{x \to 0} 4 x) \cdot lim_{x \to 0} \sin (4 x)}$

Step 5.1.3.3

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

$\frac{1}{4} \cdot \frac{0}{\cos (4 lim_{x \to 0} x) \cdot lim_{x \to 0} \sin (4 x)}$

Step 5.1.3.4

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

$\frac{1}{4} \cdot \frac{0}{\cos (4 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} 4 x)}$

Step 5.1.3.5

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

$\frac{1}{4} \cdot \frac{0}{\cos (4 lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x)}$

Step 5.1.3.6

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

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Step 5.1.3.6.1

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

$\frac{1}{4} \cdot \frac{0}{\cos (4 \cdot 0) \cdot \sin (4 lim_{x \to 0} x)}$

Step 5.1.3.6.2

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

$\frac{1}{4} \cdot \frac{0}{\cos (4 \cdot 0) \cdot \sin (4 \cdot 0)}$

Step 5.1.3.7

Simplify the answer.

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Step 5.1.3.7.1

Multiply $4$ by $0$ .

$\frac{1}{4} \cdot \frac{0}{\cos (0) \cdot \sin (4 \cdot 0)}$

Step 5.1.3.7.2

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

$\frac{1}{4} \cdot \frac{0}{1 \cdot \sin (4 \cdot 0)}$

Step 5.1.3.7.3

Multiply $\sin (4 \cdot 0)$ by $1$ .

$\frac{1}{4} \cdot \frac{0}{\sin (4 \cdot 0)}$

Step 5.1.3.7.4

Multiply $4$ by $0$ .

$\frac{1}{4} \cdot \frac{0}{\sin (0)}$

Step 5.1.3.7.5

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

$\frac{1}{4} \cdot \frac{0}{0}$

Step 5.1.3.7.6

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

Undefined

$\frac{1}{4} \cdot \frac{0}{0}$

Step 5.1.3.8

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

Undefined

$\frac{1}{4} \cdot \frac{0}{0}$

Step 5.1.4

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

Undefined

$\frac{1}{4} \cdot \frac{0}{0}$

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.

$lim_{x \to 0} \frac{x}{\cos (4 x) \sin (4 x)} = lim_{x \to 0} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [\cos (4 x) \sin (4 x)]}$

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

$\frac{1}{4} lim_{x \to 0} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [\cos (4 x) \sin (4 x)]}$

Step 5.3.2

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

$\frac{1}{4} lim_{x \to 0} \frac{1}{\frac{d}{d x} [\cos (4 x) \sin (4 x)]}$

Step 5.3.3

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 $f (x) = \cos (4 x)$ and $g (x) = \sin (4 x)$ .

$\frac{1}{4} lim_{x \to 0} \frac{1}{\cos (4 x) \frac{d}{d x} [\sin (4 x)] + \sin (4 x) \frac{d}{d x} [\cos (4 x)]}$

Step 5.3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 4 x$ .

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Step 5.3.4.1

To apply the Chain Rule, set $u_{1}$ as $4 x$ .

$\frac{1}{4} lim_{x \to 0} \frac{1}{\cos (4 x) \frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [4 x] + \sin (4 x) \frac{d}{d x} [\cos (4 x)]}$

Step 5.3.4.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$\frac{1}{4} lim_{x \to 0} \frac{1}{\cos (4 x) \cos (u_{1}) \frac{d}{d x} [4 x] + \sin (4 x) \frac{d}{d x} [\cos (4 x)]}$

Step 5.3.4.3

Replace all occurrences of $u_{1}$ with $4 x$ .

$\frac{1}{4} lim_{x \to 0} \frac{1}{\cos (4 x) \cos (4 x) \frac{d}{d x} [4 x] + \sin (4 x) \frac{d}{d x} [\cos (4 x)]}$

Step 5.3.5

Raise $\cos (4 x)$ to the power of $1$ .

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

Step 5.3.6

Raise $\cos (4 x)$ to the power of $1$ .

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

Step 5.3.7

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

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

Step 5.3.8

Add $1$ and $1$ .

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

Step 5.3.9

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

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

Step 5.3.10

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

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

Step 5.3.11

Multiply $4$ by $1$ .

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

Step 5.3.12

Move $4$ to the left of $\cos^{2} (4 x)$ .

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

Step 5.3.13

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 4 x$ .

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Step 5.3.13.1

To apply the Chain Rule, set $u_{2}$ as $4 x$ .

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

Step 5.3.13.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

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

Step 5.3.13.3

Replace all occurrences of $u_{2}$ with $4 x$ .

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

Step 5.3.14

Raise $\sin (4 x)$ to the power of $1$ .

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

Step 5.3.15

Raise $\sin (4 x)$ to the power of $1$ .

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

Step 5.3.16

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

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

Step 5.3.17

Add $1$ and $1$ .

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

Step 5.3.18

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

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

Step 5.3.19

Multiply $4$ by $- 1$ .

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

Step 5.3.20

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

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

Step 5.3.21

Multiply $- 4$ by $1$ .

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

Step 6

Evaluate the limit.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Move the exponent $2$ from $\cos^{2} (4 x)$ outside the limit using the Limits Power Rule.

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

Step 6.6

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

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

Step 6.7

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

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

Step 6.8

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

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

Step 6.9

Move the exponent $2$ from $\sin^{2} (4 x)$ outside the limit using the Limits Power Rule.

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

Step 6.10

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

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

Step 6.11

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

$\frac{1}{4} \cdot \frac{1}{4 \cos^{2} (4 \cdot 0) - 4 \sin^{2} (4 \cdot 0)}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the denominator.

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Step 8.1.1

Multiply $4$ by $0$ .

$\frac{1}{4} \cdot \frac{1}{4 \cos^{2} (0) - 4 \sin^{2} (4 \cdot 0)}$

Step 8.1.2

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

$\frac{1}{4} \cdot \frac{1}{4 \cdot 1^{2} - 4 \sin^{2} (4 \cdot 0)}$

Step 8.1.3

One to any power is one.

$\frac{1}{4} \cdot \frac{1}{4 \cdot 1 - 4 \sin^{2} (4 \cdot 0)}$

Step 8.1.4

Multiply $4$ by $1$ .

$\frac{1}{4} \cdot \frac{1}{4 - 4 \sin^{2} (4 \cdot 0)}$

Step 8.1.5

Multiply $4$ by $0$ .

$\frac{1}{4} \cdot \frac{1}{4 - 4 \sin^{2} (0)}$

Step 8.1.6

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

$\frac{1}{4} \cdot \frac{1}{4 - 4 \cdot 0^{2}}$

Step 8.1.7

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

$\frac{1}{4} \cdot \frac{1}{4 - 4 \cdot 0}$

Step 8.1.8

Multiply $- 4$ by $0$ .

$\frac{1}{4} \cdot \frac{1}{4 + 0}$

Step 8.1.9

Add $4$ and $0$ .

$\frac{1}{4} \cdot \frac{1}{4}$

Step 8.2

Multiply $\frac{1}{4} \cdot \frac{1}{4}$ .

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Step 8.2.1

Multiply $\frac{1}{4}$ by $\frac{1}{4}$ .

$\frac{1}{4 \cdot 4}$

Step 8.2.2

Multiply $4$ by $4$ .

$\frac{1}{16}$