Calculus Examples

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

Step 1

Find where the expression $\frac{1}{e^{x} - 1}$ is undefined.

$x = 0$

Step 2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{e^{x} - 1}$ approaches $0$ .

$0$

Step 3

Evaluate $lim_{x \to - \infty} \frac{1}{e^{x} - 1}$ to find the horizontal asymptote.

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Step 3.1

Evaluate the limit.

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Step 3.1.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- \infty$ .

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

Step 3.1.2

Evaluate the limit of $1$ which is constant as $x$ approaches $- \infty$ .

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

Step 3.1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- \infty$ .

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

Step 3.2

Since the exponent $x$ approaches $- \infty$ , the quantity $e^{x}$ approaches $0$ .

$\frac{1}{0 - lim_{x \to - \infty} 1}$

Step 3.3

Evaluate the limit.

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Step 3.3.1

Evaluate the limit of $1$ which is constant as $x$ approaches $- \infty$ .

$\frac{1}{0 - 1 \cdot 1}$

Step 3.3.2

Simplify the answer.

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Step 3.3.2.1

Simplify the denominator.

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Step 3.3.2.1.1

Multiply $- 1$ by $1$ .

$\frac{1}{0 - 1}$

Step 3.3.2.1.2

Subtract $1$ from $0$ .

$\frac{1}{- 1}$

Step 3.3.2.2

Divide $1$ by $- 1$ .

$- 1$

Step 4

List the horizontal asymptotes:

$y = 0, - 1$

Step 5

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 0, - 1$

No Oblique Asymptotes

Step 7