Calculus Examples

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

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 $0$ .

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

Step 1.2

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

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

Step 1.3

Move the limit into the exponent.

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Anything raised to $0$ is $1$ .

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

Step 3.1.2

Multiply $- 1$ by $1$ .

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

Step 3.1.3

Subtract $1$ from $1$ .

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

Rewrite $1$ as $1^{2}$ .

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

Step 3.2.2

Rewrite $e^{2 x}$ as ${(e^{x})}^{2}$ .

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

Step 3.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = e^{x}$ .

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

Step 3.3

Divide $0$ by $(1 + e^{x}) (1 - e^{x})$ .

$0$