Calculus Examples

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

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.

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

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 $0$ .

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

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

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

Step 1.1.2.2

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

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

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

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

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

Step 1.1.2.3.1.2

Multiply $- 1$ by $1$ .

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

Step 1.1.2.3.2

Subtract $1$ from $1$ .

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

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

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

Step 1.1.3.1.2

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

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

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

Step 1.1.3.3.2

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

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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Step 1.3.4.1

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

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

Step 1.3.4.2

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

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

Step 1.3.4.3

Multiply $- 1$ by $- 1$ .

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

Step 1.3.4.4

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

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

Step 1.3.5

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

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

Step 1.3.6

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 1.3.6.1

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

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

Step 1.3.6.2

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

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

Step 1.3.6.3

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

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

Step 1.3.7

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

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

Step 1.3.8

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

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

Step 2

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

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

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.

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

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

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

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

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

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

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

Step 3.1.3.2

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

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

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

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 $x$ .

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

Step 3.1.3.5.2

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

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

Step 3.1.3.6

Simplify the answer.

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Step 3.1.3.6.1

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

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

Step 3.1.3.6.2

One to any power is one.

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

Step 3.1.3.6.3

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

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

Step 3.1.3.6.4

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

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

Step 3.1.3.6.5

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

Undefined

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

Step 3.1.3.7

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

Undefined

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

Step 3.1.4

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

Undefined

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

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

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Step 3.3.5.1

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

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

Step 3.3.5.2

Add $2$ and $2$ .

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

Step 3.3.6

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 3.3.6.1

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

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

Step 3.3.6.2

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

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

Step 3.3.6.3

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

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

Step 3.3.7

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

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

Step 3.3.8

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

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

Step 3.3.9

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

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

Step 3.3.10

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

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

Step 3.3.11

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

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

Step 3.3.12

Add $1$ and $1$ .

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

Step 3.3.13

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

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

Step 3.3.14

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

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

Step 3.3.15

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

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

Step 3.3.16

Add $1$ and $1$ .

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

Step 3.3.17

Reorder terms.

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

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 $0$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

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

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

Step 4.9

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

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

Step 4.10

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

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

Step 4.11

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

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

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 $x$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 6

Simplify the answer.

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Step 6.1

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

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

Step 6.2

Simplify the denominator.

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Step 6.2.1

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

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

Step 6.2.2

One to any power is one.

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

Step 6.2.3

Multiply $2$ by $1$ .

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

Step 6.2.4

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

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

Step 6.2.5

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

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

Step 6.2.6

Multiply $2$ by $0$ .

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

Step 6.2.7

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

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

Step 6.2.8

One to any power is one.

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

Step 6.2.9

Add $0$ and $1$ .

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

Step 6.3

Cancel the common factor of $1$ .

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Step 6.3.1

Cancel the common factor.

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

Step 6.3.2

Rewrite the expression.

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

Step 6.4

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

$\frac{1}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$