Precalculus Examples

$lim_{x \to 0} \frac{1 - \cos (2 x)}{x \tan (2 x)}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{x \to 0} 1 - \cos (2 x)}{lim_{x \to 0} x \tan (2 x)}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

$\frac{lim_{x \to 0} 1 - lim_{x \to 0} \cos (2 x)}{lim_{x \to 0} x \tan (2 x)}$

Step 1.1.2.1.2

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

$\frac{1 - lim_{x \to 0} \cos (2 x)}{lim_{x \to 0} x \tan (2 x)}$

Step 1.1.2.1.3

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

$\frac{1 - \cos (lim_{x \to 0} 2 x)}{lim_{x \to 0} x \tan (2 x)}$

Step 1.1.2.1.4

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

$\frac{1 - \cos (2 lim_{x \to 0} x)}{lim_{x \to 0} x \tan (2 x)}$

Step 1.1.2.2

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

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

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Multiply $2$ by $0$ .

$\frac{1 - \cos (0)}{lim_{x \to 0} x \tan (2 x)}$

Step 1.1.2.3.1.2

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

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

Step 1.1.2.3.1.3

Multiply $- 1$ by $1$ .

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

Step 1.1.2.3.2

Subtract $1$ from $1$ .

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

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

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Step 1.1.3.4.1

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

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

Step 1.1.3.4.2

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

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

Step 1.1.3.5

Simplify the answer.

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Step 1.1.3.5.1

Multiply $2$ by $0$ .

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

Step 1.1.3.5.2

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

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

Step 1.1.3.5.3

Multiply $0$ by $0$ .

$\frac{0}{0}$

Step 1.1.3.5.4

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

Undefined

$\frac{0}{0}$

Step 1.1.3.6

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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Step 1.3.4.1

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

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

Step 1.3.4.2

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 $g (x) = 2 x$ .

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Step 1.3.4.2.1

To apply the Chain Rule, set $u$ as $2 x$ .

$lim_{x \to 0} \frac{0 - (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [2 x])}{\frac{d}{d x} [x \tan (2 x)]}$

Step 1.3.4.2.2

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

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

Step 1.3.4.2.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 1.3.4.3

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

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

Step 1.3.4.4

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

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

Step 1.3.4.5

Multiply $2$ by $1$ .

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

Step 1.3.4.6

Multiply $2$ by $- 1$ .

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

Step 1.3.4.7

Multiply $- 2$ by $- 1$ .

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

Step 1.3.5

Add $0$ and $2 \sin (2 x)$ .

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

Step 1.3.6

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 $g (x) = \tan (2 x)$ .

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

Step 1.3.7

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

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Step 1.3.7.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 1.3.7.2

The derivative of $\tan (u)$ with respect to $u$ is $\sec^{2} (u)$ .

$lim_{x \to 0} \frac{2 \sin (2 x)}{x (\sec^{2} (u) \frac{d}{d x} [2 x]) + \tan (2 x) \frac{d}{d x} [x]}$

Step 1.3.7.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 1.3.8

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

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

Step 1.3.9

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

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

Step 1.3.10

Multiply $2$ by $1$ .

$lim_{x \to 0} \frac{2 \sin (2 x)}{x (\sec^{2} (2 x) \cdot 2) + \tan (2 x) \frac{d}{d x} [x]}$

Step 1.3.11

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

$lim_{x \to 0} \frac{2 \sin (2 x)}{x (2 \cdot \sec^{2} (2 x)) + \tan (2 x) \frac{d}{d x} [x]}$

Step 1.3.12

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

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

Step 1.3.13

Multiply $\tan (2 x)$ by $1$ .

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

Step 1.3.14

Reorder terms.

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

Step 2

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

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

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Evaluate the limit.

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Step 3.1.2.1.1

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

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

Step 3.1.2.1.2

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

Simplify the answer.

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Step 3.1.2.3.1

Multiply $2$ by $0$ .

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

Step 3.1.2.3.2

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

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

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

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

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

Step 3.1.3.2

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

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

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

Step 3.1.3.5

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

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

Step 3.1.3.6

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

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

Step 3.1.3.7

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

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

Step 3.1.3.8

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

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

Step 3.1.3.9

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

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Step 3.1.3.9.1

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

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

Step 3.1.3.9.2

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

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

Step 3.1.3.9.3

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

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

Step 3.1.3.10

Simplify the answer.

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Step 3.1.3.10.1

Simplify each term.

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Step 3.1.3.10.1.1

Multiply $2$ by $0$ .

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

Step 3.1.3.10.1.2

Multiply $2$ by $0$ .

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

Step 3.1.3.10.1.3

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

$2 \frac{0}{0 \cdot 1^{2} + \tan (2 \cdot 0)}$

Step 3.1.3.10.1.4

One to any power is one.

$2 \frac{0}{0 \cdot 1 + \tan (2 \cdot 0)}$

Step 3.1.3.10.1.5

Multiply $0$ by $1$ .

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

Step 3.1.3.10.1.6

Multiply $2$ by $0$ .

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

Step 3.1.3.10.1.7

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

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

Step 3.1.3.10.2

Add $0$ and $0$ .

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

Step 3.1.3.10.3

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

Undefined

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

Step 3.1.3.11

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

Undefined

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

Step 3.1.4

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

Undefined

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

Step 3.2

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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 $g (x) = 2 x$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u$ as $2 x$ .

$2 lim_{x \to 0} \frac{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [2 x]}{\frac{d}{d x} [2 x \sec^{2} (2 x) + \tan (2 x)]}$

Step 3.3.2.2

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

$2 lim_{x \to 0} \frac{\cos (u) \frac{d}{d x} [2 x]}{\frac{d}{d x} [2 x \sec^{2} (2 x) + \tan (2 x)]}$

Step 3.3.2.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 3.3.3

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

$2 lim_{x \to 0} \frac{\cos (2 x) \cdot 2 \frac{d}{d x} [x]}{\frac{d}{d x} [2 x \sec^{2} (2 x) + \tan (2 x)]}$

Step 3.3.4

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

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

Step 3.3.5

Multiply $2$ by $1$ .

$2 lim_{x \to 0} \frac{\cos (2 x) \cdot 2}{\frac{d}{d x} [2 x \sec^{2} (2 x) + \tan (2 x)]}$

Step 3.3.6

Move $2$ to the left of $\cos (2 x)$ .

$2 lim_{x \to 0} \frac{2 \cdot \cos (2 x)}{\frac{d}{d x} [2 x \sec^{2} (2 x) + \tan (2 x)]}$

Step 3.3.7

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

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

Step 3.3.8

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

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Step 3.3.8.1

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

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

Step 3.3.8.2

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 $g (x) = \sec^{2} (2 x)$ .

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

Step 3.3.8.3

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 $g (x) = \sec (2 x)$ .

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Step 3.3.8.3.1

To apply the Chain Rule, set $u_{1}$ as $\sec (2 x)$ .

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

Step 3.3.8.3.2

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

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

Step 3.3.8.3.3

Replace all occurrences of $u_{1}$ with $\sec (2 x)$ .

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{2 (x \cdot 2 \sec (2 x) \frac{d}{d x} [\sec (2 x)] + \sec^{2} (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\tan (2 x)]}$

Step 3.3.8.4

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

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Step 3.3.8.4.1

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

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{2 (x \cdot 2 \sec (2 x) \frac{d}{d u_{2}} [\sec (u_{2})] \frac{d}{d x} [2 x] + \sec^{2} (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\tan (2 x)]}$

Step 3.3.8.4.2

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

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{2 (x \cdot 2 \sec (2 x) \sec (u_{2}) \tan (u_{2}) \frac{d}{d x} [2 x] + \sec^{2} (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\tan (2 x)]}$

Step 3.3.8.4.3

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

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{2 (x \cdot 2 \sec (2 x) \sec (2 x) \tan (2 x) \frac{d}{d x} [2 x] + \sec^{2} (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\tan (2 x)]}$

Step 3.3.8.5

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

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{2 (x \cdot 2 \sec (2 x) \sec (2 x) \tan (2 x) \cdot 2 \frac{d}{d x} [x] + \sec^{2} (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\tan (2 x)]}$

Step 3.3.8.6

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

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

Step 3.3.8.7

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

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

Step 3.3.8.8

Multiply $2$ by $1$ .

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

Step 3.3.8.9

Move $2$ to the left of $\sec (2 x) \tan (2 x)$ .

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

Step 3.3.8.10

Multiply $2$ by $2$ .

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

Step 3.3.8.11

Raise $\sec (2 x)$ to the power of $1$ .

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

Step 3.3.8.12

Raise $\sec (2 x)$ to the power of $1$ .

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

Step 3.3.8.13

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

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

Step 3.3.8.14

Add $1$ and $1$ .

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

Step 3.3.8.15

Multiply $\sec^{2} (2 x)$ by $1$ .

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

Step 3.3.9

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

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Step 3.3.9.1

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

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Step 3.3.9.1.1

To apply the Chain Rule, set $u_{3}$ as $2 x$ .

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{2 (x \cdot 4 \sec^{2} (2 x) \tan (2 x) + \sec^{2} (2 x)) + \frac{d}{d u_{3}} [\tan (u_{3})] \frac{d}{d x} [2 x]}$

Step 3.3.9.1.2

The derivative of $\tan (u_{3})$ with respect to $u_{3}$ is $\sec^{2} (u_{3})$ .

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{2 (x \cdot 4 \sec^{2} (2 x) \tan (2 x) + \sec^{2} (2 x)) + \sec^{2} (u_{3}) \frac{d}{d x} [2 x]}$

Step 3.3.9.1.3

Replace all occurrences of $u_{3}$ with $2 x$ .

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

Step 3.3.9.2

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

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

Step 3.3.9.3

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

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

Step 3.3.9.4

Multiply $2$ by $1$ .

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{2 (x \cdot 4 \sec^{2} (2 x) \tan (2 x) + \sec^{2} (2 x)) + \sec^{2} (2 x) \cdot 2}$

Step 3.3.9.5

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

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{2 (x \cdot 4 \sec^{2} (2 x) \tan (2 x) + \sec^{2} (2 x)) + 2 \sec^{2} (2 x)}$

Step 3.3.10

Simplify.

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Step 3.3.10.1

Apply the distributive property.

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{2 x \cdot 4 \sec^{2} (2 x) \tan (2 x) + 2 \sec^{2} (2 x) + 2 \sec^{2} (2 x)}$

Step 3.3.10.2

Combine terms.

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Step 3.3.10.2.1

Multiply $4$ by $2$ .

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{8 x \sec^{2} (2 x) \tan (2 x) + 2 \sec^{2} (2 x) + 2 \sec^{2} (2 x)}$

Step 3.3.10.2.2

Add $2 \sec^{2} (2 x)$ and $2 \sec^{2} (2 x)$ .

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

Step 3.3.10.3

Simplify each term.

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Step 3.3.10.3.1

Rewrite $\sec (2 x)$ in terms of sines and cosines.

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

Step 3.3.10.3.2

Apply the product rule to $\frac{1}{\cos (2 x)}$ .

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

Step 3.3.10.3.3

One to any power is one.

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

Step 3.3.10.3.4

Multiply $8 x \frac{1}{\cos^{2} (2 x)}$ .

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Step 3.3.10.3.4.1

Combine $8$ and $\frac{1}{\cos^{2} (2 x)}$ .

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

Step 3.3.10.3.4.2

Combine $x$ and $\frac{8}{\cos^{2} (2 x)}$ .

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

Step 3.3.10.3.5

Move $8$ to the left of $x$ .

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

Step 3.3.10.3.6

Rewrite $\tan (2 x)$ in terms of sines and cosines.

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

Step 3.3.10.3.7

Combine.

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

Step 3.3.10.3.8

Multiply $\cos^{2} (2 x)$ by $\cos (2 x)$ by adding the exponents.

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Step 3.3.10.3.8.1

Multiply $\cos^{2} (2 x)$ by $\cos (2 x)$ .

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Step 3.3.10.3.8.1.1

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 3.3.10.3.8.1.2

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

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

Step 3.3.10.3.8.2

Add $2$ and $1$ .

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{\frac{8 x \sin (2 x)}{\cos^{3} (2 x)} + 4 \sec^{2} (2 x)}$

Step 3.3.10.3.9

Rewrite $\sec (2 x)$ in terms of sines and cosines.

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{\frac{8 x \sin (2 x)}{\cos^{3} (2 x)} + 4 {(\frac{1}{\cos (2 x)})}^{2}}$

Step 3.3.10.3.10

Apply the product rule to $\frac{1}{\cos (2 x)}$ .

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{\frac{8 x \sin (2 x)}{\cos^{3} (2 x)} + 4 \frac{1^{2}}{\cos^{2} (2 x)}}$

Step 3.3.10.3.11

One to any power is one.

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{\frac{8 x \sin (2 x)}{\cos^{3} (2 x)} + 4 \frac{1}{\cos^{2} (2 x)}}$

Step 3.3.10.3.12

Combine $4$ and $\frac{1}{\cos^{2} (2 x)}$ .

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{\frac{8 x \sin (2 x)}{\cos^{3} (2 x)} + \frac{4}{\cos^{2} (2 x)}}$

Step 3.4

Combine terms.

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Step 3.4.1

To write $\frac{4}{\cos^{2} (2 x)}$ as a fraction with a common denominator, multiply by $\frac{\cos (2 x)}{\cos (2 x)}$ .

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{\frac{8 x \sin (2 x)}{\cos^{3} (2 x)} + \frac{4}{\cos^{2} (2 x)} \cdot \frac{\cos (2 x)}{\cos (2 x)}}$

Step 3.4.2

Write each expression with a common denominator of $\cos^{3} (2 x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.2.1

Multiply $\frac{4}{\cos^{2} (2 x)}$ by $\frac{\cos (2 x)}{\cos (2 x)}$ .

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{\frac{8 x \sin (2 x)}{\cos^{3} (2 x)} + \frac{4 \cos (2 x)}{\cos^{2} (2 x) \cos (2 x)}}$

Step 3.4.2.2

Multiply $\cos^{2} (2 x)$ by $\cos (2 x)$ by adding the exponents.

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Step 3.4.2.2.1

Multiply $\cos^{2} (2 x)$ by $\cos (2 x)$ .

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Step 3.4.2.2.1.1

Raise $\cos (2 x)$ to the power of $1$ .

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{\frac{8 x \sin (2 x)}{\cos^{3} (2 x)} + \frac{4 \cos (2 x)}{\cos^{2} (2 x) \cos^{1} (2 x)}}$

Step 3.4.2.2.1.2

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

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{\frac{8 x \sin (2 x)}{\cos^{3} (2 x)} + \frac{4 \cos (2 x)}{{\cos (2 x)}^{2 + 1}}}$

Step 3.4.2.2.2

Add $2$ and $1$ .

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{\frac{8 x \sin (2 x)}{\cos^{3} (2 x)} + \frac{4 \cos (2 x)}{\cos^{3} (2 x)}}$

Step 3.4.3

Combine the numerators over the common denominator.

$2 lim_{x \to 0} \frac{2 \cos (2 x)}{\frac{8 x \sin (2 x) + 4 \cos (2 x)}{\cos^{3} (2 x)}}$

Step 4

Evaluate the limit.

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Step 4.1

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

$2 \cdot 2 lim_{x \to 0} \frac{\cos (2 x)}{\frac{8 x \sin (2 x) + 4 \cos (2 x)}{\cos^{3} (2 x)}}$

Step 4.2

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

$2 \cdot 2 \frac{lim_{x \to 0} \cos (2 x)}{lim_{x \to 0} \frac{8 x \sin (2 x) + 4 \cos (2 x)}{\cos^{3} (2 x)}}$

Step 4.3

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

$2 \cdot 2 \frac{\cos (lim_{x \to 0} 2 x)}{lim_{x \to 0} \frac{8 x \sin (2 x) + 4 \cos (2 x)}{\cos^{3} (2 x)}}$

Step 4.4

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

$2 \cdot 2 \frac{\cos (2 lim_{x \to 0} x)}{lim_{x \to 0} \frac{8 x \sin (2 x) + 4 \cos (2 x)}{\cos^{3} (2 x)}}$

Step 4.5

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

$2 \cdot 2 \frac{\cos (2 lim_{x \to 0} x)}{\frac{lim_{x \to 0} 8 x \sin (2 x) + 4 \cos (2 x)}{lim_{x \to 0} \cos^{3} (2 x)}}$

Step 4.6

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

$2 \cdot 2 \frac{\cos (2 lim_{x \to 0} x)}{\frac{lim_{x \to 0} 8 x \sin (2 x) + lim_{x \to 0} 4 \cos (2 x)}{lim_{x \to 0} \cos^{3} (2 x)}}$

Step 4.7

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

$2 \cdot 2 \frac{\cos (2 lim_{x \to 0} x)}{\frac{8 lim_{x \to 0} x \sin (2 x) + lim_{x \to 0} 4 \cos (2 x)}{lim_{x \to 0} \cos^{3} (2 x)}}$

Step 4.8

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

$2 \cdot 2 \frac{\cos (2 lim_{x \to 0} x)}{\frac{8 lim_{x \to 0} x \cdot lim_{x \to 0} \sin (2 x) + lim_{x \to 0} 4 \cos (2 x)}{lim_{x \to 0} \cos^{3} (2 x)}}$

Step 4.9

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

$2 \cdot 2 \frac{\cos (2 lim_{x \to 0} x)}{\frac{8 lim_{x \to 0} x \cdot \sin (lim_{x \to 0} 2 x) + lim_{x \to 0} 4 \cos (2 x)}{lim_{x \to 0} \cos^{3} (2 x)}}$

Step 4.10

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

$2 \cdot 2 \frac{\cos (2 lim_{x \to 0} x)}{\frac{8 lim_{x \to 0} x \cdot \sin (2 lim_{x \to 0} x) + lim_{x \to 0} 4 \cos (2 x)}{lim_{x \to 0} \cos^{3} (2 x)}}$

Step 4.11

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

$2 \cdot 2 \frac{\cos (2 lim_{x \to 0} x)}{\frac{8 lim_{x \to 0} x \cdot \sin (2 lim_{x \to 0} x) + 4 lim_{x \to 0} \cos (2 x)}{lim_{x \to 0} \cos^{3} (2 x)}}$

Step 4.12

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

$2 \cdot 2 \frac{\cos (2 lim_{x \to 0} x)}{\frac{8 lim_{x \to 0} x \cdot \sin (2 lim_{x \to 0} x) + 4 \cos (lim_{x \to 0} 2 x)}{lim_{x \to 0} \cos^{3} (2 x)}}$

Step 4.13

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

$2 \cdot 2 \frac{\cos (2 lim_{x \to 0} x)}{\frac{8 lim_{x \to 0} x \cdot \sin (2 lim_{x \to 0} x) + 4 \cos (2 lim_{x \to 0} x)}{lim_{x \to 0} \cos^{3} (2 x)}}$

Step 4.14

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

$2 \cdot 2 \frac{\cos (2 lim_{x \to 0} x)}{\frac{8 lim_{x \to 0} x \cdot \sin (2 lim_{x \to 0} x) + 4 \cos (2 lim_{x \to 0} x)}{{(lim_{x \to 0} \cos (2 x))}^{3}}}$

Step 4.15

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

$2 \cdot 2 \frac{\cos (2 lim_{x \to 0} x)}{\frac{8 lim_{x \to 0} x \cdot \sin (2 lim_{x \to 0} x) + 4 \cos (2 lim_{x \to 0} x)}{\cos^{3} (lim_{x \to 0} 2 x)}}$

Step 4.16

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

$2 \cdot 2 \frac{\cos (2 lim_{x \to 0} x)}{\frac{8 lim_{x \to 0} x \cdot \sin (2 lim_{x \to 0} x) + 4 \cos (2 lim_{x \to 0} x)}{\cos^{3} (2 lim_{x \to 0} x)}}$

Step 5

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

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Step 5.1

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

$2 \cdot 2 \frac{\cos (2 \cdot 0)}{\frac{8 lim_{x \to 0} x \cdot \sin (2 lim_{x \to 0} x) + 4 \cos (2 lim_{x \to 0} x)}{\cos^{3} (2 lim_{x \to 0} x)}}$

Step 5.2

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

$2 \cdot 2 \frac{\cos (2 \cdot 0)}{\frac{8 \cdot 0 \cdot \sin (2 lim_{x \to 0} x) + 4 \cos (2 lim_{x \to 0} x)}{\cos^{3} (2 lim_{x \to 0} x)}}$

Step 5.3

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

$2 \cdot 2 \frac{\cos (2 \cdot 0)}{\frac{8 \cdot 0 \cdot \sin (2 \cdot 0) + 4 \cos (2 lim_{x \to 0} x)}{\cos^{3} (2 lim_{x \to 0} x)}}$

Step 5.4

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

$2 \cdot 2 \frac{\cos (2 \cdot 0)}{\frac{8 \cdot 0 \cdot \sin (2 \cdot 0) + 4 \cos (2 \cdot 0)}{\cos^{3} (2 lim_{x \to 0} x)}}$

Step 5.5

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

$2 \cdot 2 \frac{\cos (2 \cdot 0)}{\frac{8 \cdot 0 \cdot \sin (2 \cdot 0) + 4 \cos (2 \cdot 0)}{\cos^{3} (2 \cdot 0)}}$

Step 6

Simplify the answer.

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Step 6.1

Cancel the common factor of $2$ .

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Step 6.1.1

Factor $2$ out of $2 \cdot 2$ .

$2 (2) \frac{\cos (2 \cdot 0)}{\frac{8 \cdot 0 \cdot \sin (2 \cdot 0) + 4 \cos (2 \cdot 0)}{\cos^{3} (2 \cdot 0)}}$

Step 6.1.2

Factor $2$ out of $\frac{8 \cdot 0 \cdot \sin (2 \cdot 0) + 4 \cos (2 \cdot 0)}{\cos^{3} (2 \cdot 0)}$ .

$2 (2) \frac{\cos (2 \cdot 0)}{2 \frac{4 \cdot 0 \cdot \sin (2 \cdot 0) + 2 \cos (2 \cdot 0)}{\cos^{3} (2 \cdot 0)}}$

Step 6.1.3

Cancel the common factor.

$2 \cdot 2 \frac{\cos (2 \cdot 0)}{2 \frac{4 \cdot 0 \cdot \sin (2 \cdot 0) + 2 \cos (2 \cdot 0)}{\cos^{3} (2 \cdot 0)}}$

Step 6.1.4

Rewrite the expression.

$2 \frac{\cos (2 \cdot 0)}{\frac{4 \cdot 0 \cdot \sin (2 \cdot 0) + 2 \cos (2 \cdot 0)}{\cos^{3} (2 \cdot 0)}}$

Step 6.2

Multiply $2$ by $0$ .

$2 \frac{\cos (0)}{\frac{4 \cdot 0 \cdot \sin (2 \cdot 0) + 2 \cos (2 \cdot 0)}{\cos^{3} (2 \cdot 0)}}$

Step 6.3

Multiply $4$ by $0$ .

$2 \frac{\cos (0)}{\frac{0 \cdot \sin (2 \cdot 0) + 2 \cos (2 \cdot 0)}{\cos^{3} (2 \cdot 0)}}$

Step 6.4

Multiply $2$ by $0$ .

$2 \frac{\cos (0)}{\frac{0 \cdot \sin (0) + 2 \cos (2 \cdot 0)}{\cos^{3} (2 \cdot 0)}}$

Step 6.5

Multiply $0$ by $\sin (0)$ .

$2 \frac{\cos (0)}{\frac{0 + 2 \cos (2 \cdot 0)}{\cos^{3} (2 \cdot 0)}}$

Step 6.6

Multiply $2$ by $0$ .

$2 \frac{\cos (0)}{\frac{0 + 2 \cos (0)}{\cos^{3} (2 \cdot 0)}}$

Step 6.7

Multiply $2$ by $0$ .

$2 \frac{\cos (0)}{\frac{0 + 2 \cos (0)}{\cos^{3} (0)}}$

Step 6.8

Combine $2$ and $\frac{\cos (0)}{\frac{0 + 2 \cos (0)}{\cos^{3} (0)}}$ .

$\frac{2 \cos (0)}{\frac{0 + 2 \cos (0)}{\cos^{3} (0)}}$

Step 6.9

Simplify the numerator.

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Step 6.9.1

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

$\frac{2 \cos (0)}{\frac{0 + 2 \cdot 1}{\cos^{3} (0)}}$

Step 6.9.2

Multiply $2$ by $1$ .

$\frac{2 \cos (0)}{\frac{0 + 2}{\cos^{3} (0)}}$

Step 6.9.3

Add $0$ and $2$ .

$\frac{2 \cos (0)}{\frac{2}{\cos^{3} (0)}}$

Step 6.10

Simplify the denominator.

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Step 6.10.1

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

$\frac{2 \cos (0)}{\frac{2}{1^{3}}}$

Step 6.10.2

One to any power is one.

$\frac{2 \cos (0)}{\frac{2}{1}}$

Step 6.11

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

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

Step 6.12

Divide $2$ by $1$ .

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

Step 6.13

Multiply $2$ by $1$ .

$\frac{2}{2}$

Step 6.14

Cancel the common factor of $2$ .

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Step 6.14.1

Cancel the common factor.

$\frac{2}{2}$

Step 6.14.2

Rewrite the expression.

$1$