Calculus Examples

$y = \frac{e^{x} + 1}{1 - e^{x}}$

Step 1

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

$x = 0$

Step 2

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

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Step 2.1

Apply L'Hospital's rule.

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Step 2.1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 2.1.1.1

Take the limit of the numerator and the limit of the denominator.

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

Step 2.1.1.2

Evaluate the limit of the numerator.

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Step 2.1.1.2.1

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

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

Step 2.1.1.2.2

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

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

Step 2.1.1.2.3

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

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

Step 2.1.1.2.4

Infinity plus or minus a number is infinity.

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

Step 2.1.1.3

Evaluate the limit of the denominator.

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Step 2.1.1.3.1

Evaluate the limit.

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Step 2.1.1.3.1.1

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

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

Step 2.1.1.3.1.2

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

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

Step 2.1.1.3.2

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

$\frac{\infty}{1 - \infty}$

Step 2.1.1.3.3

Simplify the answer.

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Step 2.1.1.3.3.1

A non-zero constant times infinity is infinity.

$\frac{\infty}{1 + - \infty}$

Step 2.1.1.3.3.2

Infinity plus or minus a number is infinity.

$\frac{\infty}{- \infty}$

Step 2.1.1.3.3.3

Infinity divided by infinity is undefined.

Undefined

$\frac{\infty}{- \infty}$

Step 2.1.1.3.4

Infinity divided by infinity is undefined.

Undefined

$\frac{\infty}{- \infty}$

Step 2.1.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{\infty}{- \infty}$

Step 2.1.2

Since $\frac{\infty}{- \infty}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to \infty} \frac{e^{x} + 1}{1 - e^{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x} + 1]}{\frac{d}{d x} [1 - e^{x}]}$

Step 2.1.3

Find the derivative of the numerator and denominator.

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Step 2.1.3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x} + 1]}{\frac{d}{d x} [1 - e^{x}]}$

Step 2.1.3.2

By the Sum Rule, the derivative of $e^{x} + 1$ with respect to $x$ is $\frac{d}{d x} [e^{x}] + \frac{d}{d x} [1]$ .

$lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x}] + \frac{d}{d x} [1]}{\frac{d}{d x} [1 - e^{x}]}$

Step 2.1.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$lim_{x \to \infty} \frac{e^{x} + \frac{d}{d x} [1]}{\frac{d}{d x} [1 - e^{x}]}$

Step 2.1.3.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$lim_{x \to \infty} \frac{e^{x} + 0}{\frac{d}{d x} [1 - e^{x}]}$

Step 2.1.3.5

Add $e^{x}$ and $0$ .

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

Step 2.1.3.6

By the Sum Rule, the derivative of $1 - e^{x}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- e^{x}]$ .

$lim_{x \to \infty} \frac{e^{x}}{\frac{d}{d x} [1] + \frac{d}{d x} [- e^{x}]}$

Step 2.1.3.7

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$lim_{x \to \infty} \frac{e^{x}}{0 + \frac{d}{d x} [- e^{x}]}$

Step 2.1.3.8

Evaluate $\frac{d}{d x} [- e^{x}]$ .

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Step 2.1.3.8.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- e^{x}$ with respect to $x$ is $- \frac{d}{d x} [e^{x}]$ .

$lim_{x \to \infty} \frac{e^{x}}{0 - \frac{d}{d x} [e^{x}]}$

Step 2.1.3.8.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

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

Step 2.1.3.9

Subtract $e^{x}$ from $0$ .

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

Step 2.1.4

Cancel the common factor of $e^{x}$ .

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Step 2.1.4.1

Cancel the common factor.

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

Step 2.1.4.2

Rewrite the expression.

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

Step 2.1.4.3

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$lim_{x \to \infty} - 1 \cdot 1$

Step 2.2

Evaluate the limit.

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Step 2.2.1

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

$- 1 \cdot 1$

Step 2.2.2

Multiply $- 1$ by $1$ .

$- 1$

Step 3

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

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

Step 3.1.2

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

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

Step 3.2

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

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

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 $- \infty$ .

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

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

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

Step 3.5

Simplify the answer.

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Step 3.5.1

Add $0$ and $1$ .

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

Step 3.5.2

Simplify the denominator.

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Step 3.5.2.1

Multiply $- 1$ by $0$ .

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

Step 3.5.2.2

Add $1$ and $0$ .

$\frac{1}{1}$

Step 3.5.3

Divide $1$ by $1$ .

$1$

Step 4

List the horizontal asymptotes:

$y = - 1, 1$

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: $y = - 1, 1$

No Oblique Asymptotes

Step 7