Calculus Examples

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

Step 1

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

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

Step 2

Move the limit into the exponent.

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

Step 3

Move the limit into the exponent.

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

Step 4

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

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

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 $x$ .

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

Step 5.2

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

$\frac{e^{0} - e^{0}}{\sin (2 x)}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Anything raised to $0$ is $1$ .

$\frac{1 - e^{0}}{\sin (2 x)}$

Step 6.1.2

Anything raised to $0$ is $1$ .

$\frac{1 - 1 \cdot 1}{\sin (2 x)}$

Step 6.1.3

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{\sin (2 x)}$

Step 6.1.4

Subtract $1$ from $1$ .

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

Step 6.2

Separate fractions.

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

Step 6.3

Convert from $\frac{1}{\sin (2 x)}$ to $\csc (2 x)$ .

$\frac{0}{1} \csc (2 x)$

Step 6.4

Divide $0$ by $1$ .

$0 \csc (2 x)$

Step 6.5

Multiply $0$ by $\csc (2 x)$ .

$0$