Calculus Examples

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

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.

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

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

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

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

Step 1.2.2

Move the limit into the exponent.

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

Step 1.2.3

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

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

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

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

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

Step 1.2.5

Simplify the answer.

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Step 1.2.5.1

Simplify each term.

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Step 1.2.5.1.1

Anything raised to $0$ is $1$ .

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

Step 1.2.5.1.2

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

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

Step 1.2.5.1.3

Multiply $- 1$ by $1$ .

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

Step 1.2.5.2

Subtract $1$ from $1$ .

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

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

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

Step 1.3.1.2

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

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

Step 1.3.2

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

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

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

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

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

Step 1.3.3.2

Multiply $4$ by $0$ .

$\frac{0}{0}$

Step 1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

Multiply $- 1$ by $- 1$ .

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

Step 3.4.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Move the limit into the exponent.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Anything raised to $0$ is $1$ .

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

Step 11.1.2

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

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

Step 11.1.3

Add $1$ and $0$ .

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

Step 11.2

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

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

Step 11.3

Cancel the common factor of $1$ .

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Step 11.3.1

Cancel the common factor.

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

Step 11.3.2

Rewrite the expression.

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

Step 11.4

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

$\frac{1}{4}$