Calculus Examples

$lim_{x \to 0} \frac{e^{- x} - e^{x}}{e^{- x} + 1}$

Step 1

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

$\frac{lim_{x \to 0} e^{- x} - e^{x}}{lim_{x \to 0} e^{- x} + 1}$

Step 2

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

$\frac{lim_{x \to 0} e^{- x} - lim_{x \to 0} e^{x}}{lim_{x \to 0} e^{- x} + 1}$

Step 3

Move the limit into the exponent.

$\frac{e^{lim_{x \to 0} - x} - lim_{x \to 0} e^{x}}{lim_{x \to 0} e^{- x} + 1}$

Step 4

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

$\frac{e^{- lim_{x \to 0} x} - lim_{x \to 0} e^{x}}{lim_{x \to 0} e^{- x} + 1}$

Step 5

Move the limit into the exponent.

$\frac{e^{- lim_{x \to 0} x} - e^{lim_{x \to 0} x}}{lim_{x \to 0} e^{- x} + 1}$

Step 6

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

$\frac{e^{- lim_{x \to 0} x} - e^{lim_{x \to 0} x}}{lim_{x \to 0} e^{- x} + lim_{x \to 0} 1}$

Step 7

Move the limit into the exponent.

$\frac{e^{- lim_{x \to 0} x} - e^{lim_{x \to 0} x}}{e^{lim_{x \to 0} - x} + lim_{x \to 0} 1}$

Step 8

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

$\frac{e^{- lim_{x \to 0} x} - e^{lim_{x \to 0} x}}{e^{- lim_{x \to 0} x} + lim_{x \to 0} 1}$

Step 9

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

$\frac{e^{- lim_{x \to 0} x} - e^{lim_{x \to 0} x}}{e^{- lim_{x \to 0} x} + 1}$

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 $x$ .

$\frac{e^{0} - e^{lim_{x \to 0} x}}{e^{- lim_{x \to 0} x} + 1}$

Step 10.2

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

$\frac{e^{0} - e^{0}}{e^{- lim_{x \to 0} x} + 1}$

Step 10.3

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

$\frac{e^{0} - e^{0}}{e^{0} + 1}$

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 $1$ .

$\frac{1 - e^{0}}{e^{0} + 1}$

Step 11.1.2

Anything raised to $0$ is $1$ .

$\frac{1 - 1 \cdot 1}{e^{0} + 1}$

Step 11.1.3

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{e^{0} + 1}$

Step 11.1.4

Subtract $1$ from $1$ .

$\frac{0}{e^{0} + 1}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Anything raised to $0$ is $1$ .

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

Step 11.2.2

Add $1$ and $1$ .

$\frac{0}{2}$

Step 11.3

Divide $0$ by $2$ .

$0$