Calculus Examples

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

Step 1

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

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

Step 2

Move the limit into the exponent.

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

Step 3

Move the limit into the exponent.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Anything raised to $0$ is $1$ .

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

Step 7.1.2

Anything raised to $0$ is $1$ .

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

Step 7.1.3

Multiply $- 1$ by $1$ .

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

Step 7.1.4

Multiply $- 2$ by $0$ .

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

Step 7.1.5

Add $1 - 1$ and $0$ .

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

Step 7.1.6

Subtract $1$ from $1$ .

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

Step 7.2

Divide $0$ by $x - \sin (x)$ .

$0$