Calculus Examples

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

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.

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

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 $0$ .

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

Step 1.2.2

Move the limit into the exponent.

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

Step 1.2.3

Move the limit into the exponent.

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

Step 1.2.4

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 1.2.5

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

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Step 1.2.5.1

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

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

Step 1.2.5.2

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

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

Step 1.2.6

Simplify the answer.

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Step 1.2.6.1

Simplify each term.

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Step 1.2.6.1.1

Anything raised to $0$ is $1$ .

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

Step 1.2.6.1.2

Anything raised to $0$ is $1$ .

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

Step 1.2.6.1.3

Multiply $- 1$ by $1$ .

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

Step 1.2.6.2

Subtract $1$ from $1$ .

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

$\frac{0}{0}$

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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Evaluate $\frac{d}{d x} [- e^{- x}]$ .

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Step 3.4.1

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

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

Step 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 $f (x) = e^{x}$ and $g (x) = - x$ .

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Step 3.4.2.1

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

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

Step 3.4.2.2

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

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

Step 3.4.2.3

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

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

Step 3.4.3

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

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

Step 3.4.4

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

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

Step 3.4.5

Multiply $- 1$ by $1$ .

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

Step 3.4.6

Move $- 1$ to the left of $e^{- x}$ .

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

Step 3.4.7

Rewrite $- 1 e^{- x}$ as $- e^{- x}$ .

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

Step 3.4.8

Multiply $- 1$ by $- 1$ .

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

Step 3.4.9

Multiply $e^{- x}$ by $1$ .

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

Step 3.5

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

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

Step 4

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

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

Step 5

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

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

Step 6

Move the limit into the exponent.

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

Step 7

Move the limit into the exponent.

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

Step 8

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 9

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

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

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 $x$ .

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

Step 10.2

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

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

Step 10.3

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

$\frac{e^{0} + e^{0}}{\cos (0)}$

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 $1$ .

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

Step 11.1.2

Anything raised to $0$ is $1$ .

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

Step 11.1.3

Add $1$ and $1$ .

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

Step 11.2

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

$\frac{2}{1}$

Step 11.3

Divide $2$ by $1$ .

$2$