Calculus Examples

$\frac{lim_{x \to 0} {(12)}^{x} - 13^{x}}{x}$

Step 1

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

$\frac{lim_{x \to 0} {(12)}^{x} - lim_{x \to 0} 13^{x}}{x}$

Step 2

Move the limit into the exponent.

$\frac{12^{lim_{x \to 0} x} - lim_{x \to 0} 13^{x}}{x}$

Step 3

Move the limit into the exponent.

$\frac{12^{lim_{x \to 0} x} - 13^{lim_{x \to 0} x}}{x}$

Step 4

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

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Step 4.1

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

$\frac{12^{0} - 13^{lim_{x \to 0} x}}{x}$

Step 4.2

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

$\frac{12^{0} - 13^{0}}{x}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Anything raised to $0$ is $1$ .

$\frac{1 - 13^{0}}{x}$

Step 5.1.2

Anything raised to $0$ is $1$ .

$\frac{1 - 1 \cdot 1}{x}$

Step 5.1.3

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{x}$

Step 5.1.4

Subtract $1$ from $1$ .

$\frac{0}{x}$

Step 5.2

Divide $0$ by $x$ .

$0$