Precalculus Examples

$f (x) = \frac{5}{1 + 4 e^{- 2 x}}$

Step 1

Find where the expression $\frac{5}{1 + 4 e^{- 2 x}}$ 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 $lim_{x \to \infty} \frac{5}{1 + 4 e^{- 2 x}}$ to find the horizontal asymptote.

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Step 3.1

Evaluate the limit.

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Step 3.1.1

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$5 lim_{x \to \infty} \frac{1}{1 + 4 e^{- 2 x}}$

Step 3.1.2

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

$5 \frac{lim_{x \to \infty} 1}{lim_{x \to \infty} 1 + 4 e^{- 2 x}}$

Step 3.1.3

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

$5 \frac{1}{lim_{x \to \infty} 1 + 4 e^{- 2 x}}$

Step 3.1.4

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

$5 \frac{1}{lim_{x \to \infty} 1 + lim_{x \to \infty} 4 e^{- 2 x}}$

Step 3.1.5

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

$5 \frac{1}{1 + lim_{x \to \infty} 4 e^{- 2 x}}$

Step 3.1.6

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$5 \frac{1}{1 + 4 lim_{x \to \infty} e^{- 2 x}}$

Step 3.2

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

$5 \frac{1}{1 + 4 \cdot 0}$

Step 3.3

Simplify the answer.

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Step 3.3.1

Simplify the denominator.

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Step 3.3.1.1

Multiply $4$ by $0$ .

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

Step 3.3.1.2

Add $1$ and $0$ .

$5 (\frac{1}{1})$

Step 3.3.2

Cancel the common factor of $1$ .

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Step 3.3.2.1

Cancel the common factor.

$5 (\frac{1}{1})$

Step 3.3.2.2

Rewrite the expression.

$5 \cdot 1$

Step 3.3.3

Multiply $5$ by $1$ .

$5$

Step 4

Evaluate $lim_{x \to - \infty} \frac{5}{1 + 4 e^{- 2 x}}$ to find the horizontal asymptote.

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Step 4.1

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$5 lim_{x \to - \infty} \frac{1}{1 + 4 e^{- 2 x}}$

Step 4.2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{1 + 4 e^{- 2 x}}$ approaches $0$ .

$5 \cdot 0$

Step 4.3

Multiply $5$ by $0$ .

$0$

Step 5

List the horizontal asymptotes:

$y = 5, 0$

Step 6

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 7

This is the set of all asymptotes.

No Vertical Asymptotes

Horizontal Asymptotes: $y = 5, 0$

No Oblique Asymptotes

Step 8