Calculus Examples

$\frac{lim_{x \to 0} e^{2 x} - 1}{x}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{x \to 0} e^{2 x} - lim_{x \to 0} 1}{x}$

Step 1.2

Move the limit into the exponent.

$\frac{e^{lim_{x \to 0} 2 x} - lim_{x \to 0} 1}{x}$

Step 1.3

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

$\frac{e^{2 lim_{x \to 0} x} - lim_{x \to 0} 1}{x}$

Step 1.4

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

$\frac{e^{2 lim_{x \to 0} x} - 1 \cdot 1}{x}$

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $2$ by $0$ .

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

Step 3.1.2

Anything raised to $0$ is $1$ .

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

Step 3.1.3

Multiply $- 1$ by $1$ .

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

Step 3.1.4

Subtract $1$ from $1$ .

$\frac{0}{x}$

Step 3.2

Divide $0$ by $x$ .

$0$