Calculus Examples

$lim_{x \to 0} \frac{3^{x} - 3^{- x}}{3^{x} + 3^{- x}}$

Step 1

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

$\frac{lim_{x \to 0} 3^{x} - 3^{- x}}{lim_{x \to 0} 3^{x} + 3^{- x}}$

Step 2

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

$\frac{lim_{x \to 0} 3^{x} - lim_{x \to 0} 3^{- x}}{lim_{x \to 0} 3^{x} + 3^{- x}}$

Step 3

Move the limit into the exponent.

$\frac{3^{lim_{x \to 0} x} - lim_{x \to 0} 3^{- x}}{lim_{x \to 0} 3^{x} + 3^{- x}}$

Step 4

Move the limit into the exponent.

$\frac{3^{lim_{x \to 0} x} - 3^{lim_{x \to 0} - x}}{lim_{x \to 0} 3^{x} + 3^{- x}}$

Step 5

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

$\frac{3^{lim_{x \to 0} x} - 3^{- lim_{x \to 0} x}}{lim_{x \to 0} 3^{x} + 3^{- x}}$

Step 6

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

$\frac{3^{lim_{x \to 0} x} - 3^{- lim_{x \to 0} x}}{lim_{x \to 0} 3^{x} + lim_{x \to 0} 3^{- x}}$

Step 7

Move the limit into the exponent.

$\frac{3^{lim_{x \to 0} x} - 3^{- lim_{x \to 0} x}}{3^{lim_{x \to 0} x} + lim_{x \to 0} 3^{- x}}$

Step 8

Move the limit into the exponent.

$\frac{3^{lim_{x \to 0} x} - 3^{- lim_{x \to 0} x}}{3^{lim_{x \to 0} x} + 3^{lim_{x \to 0} - x}}$

Step 9

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

$\frac{3^{lim_{x \to 0} x} - 3^{- lim_{x \to 0} x}}{3^{lim_{x \to 0} x} + 3^{- lim_{x \to 0} x}}$

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 $x$ .

$\frac{3^{0} - 3^{- lim_{x \to 0} x}}{3^{lim_{x \to 0} x} + 3^{- lim_{x \to 0} x}}$

Step 10.2

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

$\frac{3^{0} - 3^{0}}{3^{lim_{x \to 0} x} + 3^{- lim_{x \to 0} x}}$

Step 10.3

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

$\frac{3^{0} - 3^{0}}{3^{0} + 3^{- lim_{x \to 0} x}}$

Step 10.4

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

$\frac{3^{0} - 3^{0}}{3^{0} + 3^{0}}$

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 $1$ .

$\frac{1 - 3^{0}}{3^{0} + 3^{0}}$

Step 11.1.2

Anything raised to $0$ is $1$ .

$\frac{1 - 1 \cdot 1}{3^{0} + 3^{0}}$

Step 11.1.3

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{3^{0} + 3^{0}}$

Step 11.1.4

Subtract $1$ from $1$ .

$\frac{0}{3^{0} + 3^{0}}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Anything raised to $0$ is $1$ .

$\frac{0}{1 + 3^{0}}$

Step 11.2.2

Anything raised to $0$ is $1$ .

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

Step 11.2.3

Add $1$ and $1$ .

$\frac{0}{2}$

Step 11.3

Divide $0$ by $2$ .

$0$