Find the Asymptotes f(x)=20/(1+9e^(3x))

$f (x) = \frac{20}{1 + 9 e^{3 x}}$

Step 1

Find where the expression $\frac{20}{1 + 9 e^{3 x}}$ 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 $lim_{x \to \infty} \frac{20}{1 + 9 e^{3 x}}$ to find the horizontal asymptote.

Move the term $20$ outside of the limit because it is constant with respect to $x$ .

$20 lim_{x \to \infty} \frac{1}{1 + 9 e^{3 x}}$

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{1 + 9 e^{3 x}}$ approaches $0$ .

$20 \cdot 0$

Multiply $20$ by $0$ .

$0$

Step 4

Evaluate $lim_{x \to - \infty} \frac{20}{1 + 9 e^{3 x}}$ to find the horizontal asymptote.

Evaluate the limit.

Move the term $20$ outside of the limit because it is constant with respect to $x$ .

$20 lim_{x \to - \infty} \frac{1}{1 + 9 e^{3 x}}$

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

$20 \frac{lim_{x \to - \infty} 1}{lim_{x \to - \infty} 1 + 9 e^{3 x}}$

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

$20 \frac{1}{lim_{x \to - \infty} 1 + 9 e^{3 x}}$

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

$20 \frac{1}{lim_{x \to - \infty} 1 + lim_{x \to - \infty} 9 e^{3 x}}$

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

$20 \frac{1}{1 + lim_{x \to - \infty} 9 e^{3 x}}$

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$20 \frac{1}{1 + 9 lim_{x \to - \infty} e^{3 x}}$

Since the exponent $3 x$ approaches $- \infty$ , the quantity $e^{3 x}$ approaches $0$ .

$20 \frac{1}{1 + 9 \cdot 0}$

Simplify the answer.

Simplify the denominator.

Multiply $9$ by $0$ .

$20 \frac{1}{1 + 0}$

Add $1$ and $0$ .

$20 (\frac{1}{1})$

Cancel the common factor of $1$ .

Cancel the common factor.

$20 (\frac{1}{1})$

Rewrite the expression.

$20 \cdot 1$

Multiply $20$ by $1$ .

$20$

Step 5

List the horizontal asymptotes:

$y = 0, 20$

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: $y = 0, 20$

No Oblique Asymptotes

Step 8

Find the Asymptotes f(x)=20/(1+9e^(3x))

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