Calculus Examples

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

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.

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

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

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

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

Step 1.1.2.1.2

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

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

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

Step 1.1.2.3.2

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

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

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 $0$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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Step 1.1.3.3.1

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

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

Step 1.1.3.3.2

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

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

Step 1.1.3.4

Simplify the answer.

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Step 1.1.3.4.1

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

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

Step 1.1.3.4.2

Multiply $0$ by $0$ .

$\frac{0}{0}$

Step 1.1.3.4.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.5

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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Step 1.3.2.1

To apply the Chain Rule, set $u$ as $\tan (x)$ .

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

Step 1.3.2.2

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

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

Step 1.3.2.3

Replace all occurrences of $u$ with $\tan (x)$ .

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

Step 1.3.3

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

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

Step 1.3.4

Reorder the factors of $2 \tan (x) \sec^{2} (x)$ .

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

Step 1.3.5

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 $g (x) = \sin (x)$ .

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

Step 1.3.6

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

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

Step 1.3.7

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

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

Step 1.3.8

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

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

Step 2

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

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

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.

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

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

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

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

Step 3.1.2.4

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

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

Step 3.1.2.5

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

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Step 3.1.2.5.1

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

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

Step 3.1.2.5.2

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

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

Step 3.1.2.6

Simplify the answer.

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Step 3.1.2.6.1

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

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

Step 3.1.2.6.2

One to any power is one.

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

Step 3.1.2.6.3

Multiply $\tan (0)$ by $1$ .

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

Step 3.1.2.6.4

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

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

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 $0$ .

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

Step 3.1.3.2

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

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

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

Step 3.1.3.5

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

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Step 3.1.3.5.1

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

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

Step 3.1.3.5.2

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

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

Step 3.1.3.5.3

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

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

Step 3.1.3.6

Simplify the answer.

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Step 3.1.3.6.1

Simplify each term.

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Step 3.1.3.6.1.1

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

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

Step 3.1.3.6.1.2

Multiply $0$ by $1$ .

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

Step 3.1.3.6.1.3

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

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

Step 3.1.3.6.2

Add $0$ and $0$ .

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

Step 3.1.3.6.3

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

Undefined

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

Step 3.1.3.7

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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Step 3.3.4.1

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

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

Step 3.3.4.2

Add $2$ and $2$ .

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

Step 3.3.5

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

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Step 3.3.5.1

To apply the Chain Rule, set $u$ as $\sec (x)$ .

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

Step 3.3.5.2

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

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

Step 3.3.5.3

Replace all occurrences of $u$ with $\sec (x)$ .

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

Step 3.3.6

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

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

Step 3.3.7

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

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

Step 3.3.8

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

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

Step 3.3.9

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

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

Step 3.3.10

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

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

Step 3.3.11

Add $1$ and $1$ .

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

Step 3.3.12

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

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

Step 3.3.13

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

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

Step 3.3.14

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

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

Step 3.3.15

Add $1$ and $1$ .

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

Step 3.3.16

Reorder terms.

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

Step 3.3.17

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

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

Step 3.3.18

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

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Step 3.3.18.1

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 $g (x) = \cos (x)$ .

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

Step 3.3.18.2

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

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

Step 3.3.18.3

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

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

Step 3.3.18.4

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

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

Step 3.3.19

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

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

Step 3.3.20

Simplify.

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Step 3.3.20.1

Add $\cos (x)$ and $\cos (x)$ .

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

Step 3.3.20.2

Reorder terms.

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

Step 4

Evaluate the limit.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

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

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

Step 4.9

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

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

Step 4.10

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

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

Step 4.11

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

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

Step 4.12

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

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

Step 4.13

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

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

Step 4.14

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

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

Step 4.15

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

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

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 $x$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

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

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

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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

Step 6.1.2

One to any power is one.

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

Step 6.1.3

Multiply $2$ by $1$ .

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

Step 6.1.4

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

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

Step 6.1.5

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

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

Step 6.1.6

Multiply $2$ by $0$ .

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

Step 6.1.7

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

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

Step 6.1.8

One to any power is one.

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

Step 6.1.9

Add $0$ and $1$ .

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

Step 6.2

Simplify the denominator.

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Step 6.2.1

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

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

Step 6.2.2

Multiply $0$ by $0$ .

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

Step 6.2.3

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

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

Step 6.2.4

Multiply $2$ by $1$ .

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

Step 6.2.5

Add $0$ and $2$ .

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

Step 6.3

Cancel the common factor of $2$ .

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Step 6.3.1

Cancel the common factor.

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

Step 6.3.2

Rewrite the expression.

$1$