Precalculus Examples

$f (x) = \frac{2^{x} + 5}{e^{x} + 1}$

Step 1

Find where the expression $\frac{2^{x} + 5}{e^{x} + 1}$ is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

Evaluate $lim_{x \to - \infty} \frac{2^{x} + 5}{e^{x} + 1}$ 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 $- \infty$ .

$\frac{lim_{x \to - \infty} 2^{x} + 5}{lim_{x \to - \infty} e^{x} + 1}$

Step 3.1.2

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

$\frac{lim_{x \to - \infty} 2^{x} + lim_{x \to - \infty} 5}{lim_{x \to - \infty} e^{x} + 1}$

Step 3.2

Since the exponent $x$ approaches $- \infty$ , the quantity $2^{x}$ approaches $0$ .

$\frac{0 + lim_{x \to - \infty} 5}{lim_{x \to - \infty} e^{x} + 1}$

Step 3.3

Evaluate the limit.

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Step 3.3.1

Evaluate the limit of $5$ which is constant as $x$ approaches $- \infty$ .

$\frac{0 + 5}{lim_{x \to - \infty} e^{x} + 1}$

Step 3.3.2

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

$\frac{0 + 5}{lim_{x \to - \infty} e^{x} + lim_{x \to - \infty} 1}$

Step 3.4

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

$\frac{0 + 5}{0 + lim_{x \to - \infty} 1}$

Step 3.5

Evaluate the limit.

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Step 3.5.1

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

$\frac{0 + 5}{0 + 1}$

Step 3.5.2

Simplify the answer.

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Step 3.5.2.1

Add $0$ and $5$ .

$\frac{5}{0 + 1}$

Step 3.5.2.2

Add $0$ and $1$ .

$\frac{5}{1}$

Step 3.5.2.3

Divide $5$ by $1$ .

$5$

Step 4

List the horizontal asymptotes:

$y = 5$

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.

No Vertical Asymptotes

Horizontal Asymptotes: $y = 5$

No Oblique Asymptotes

Step 7