Calculus Examples

$\frac{12 e^{- 2 x}}{{(1 + 3 e^{- 2 x})}^{2}}$

Step 1

Find where the expression $\frac{12 e^{- 2 x}}{{(1 + 3 e^{- 2 x})}^{2}}$ 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 $lim_{x \to \infty} \frac{12 e^{- 2 x}}{{(1 + 3 e^{- 2 x})}^{2}}$ to find the horizontal asymptote.

Tap for more steps...

Step 3.1

Evaluate the limit.

Tap for more steps...

Step 3.1.1

Move the term $12$ outside of the limit because it is constant with respect to $x$ .

$12 lim_{x \to \infty} \frac{e^{- 2 x}}{{(1 + 3 e^{- 2 x})}^{2}}$

Step 3.1.2

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

$12 \frac{lim_{x \to \infty} e^{- 2 x}}{lim_{x \to \infty} {(1 + 3 e^{- 2 x})}^{2}}$

Step 3.2

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

$12 \frac{0}{lim_{x \to \infty} {(1 + 3 e^{- 2 x})}^{2}}$

Step 3.3

Evaluate the limit.

Tap for more steps...

Step 3.3.1

Move the exponent $2$ from ${(1 + 3 e^{- 2 x})}^{2}$ outside the limit using the Limits Power Rule.

$12 \frac{0}{{(lim_{x \to \infty} 1 + 3 e^{- 2 x})}^{2}}$

Step 3.3.2

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

$12 \frac{0}{{(lim_{x \to \infty} 1 + lim_{x \to \infty} 3 e^{- 2 x})}^{2}}$

Step 3.3.3

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

$12 \frac{0}{{(1 + lim_{x \to \infty} 3 e^{- 2 x})}^{2}}$

Step 3.3.4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$12 \frac{0}{{(1 + 3 lim_{x \to \infty} e^{- 2 x})}^{2}}$

Step 3.4

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

$12 \frac{0}{{(1 + 3 \cdot 0)}^{2}}$

Step 3.5

Simplify the answer.

Tap for more steps...

Step 3.5.1

Simplify the denominator.

Tap for more steps...

Step 3.5.1.1

Multiply $3$ by $0$ .

$12 \frac{0}{{(1 + 0)}^{2}}$

Step 3.5.1.2

Add $1$ and $0$ .

$12 \frac{0}{1^{2}}$

Step 3.5.1.3

One to any power is one.

$12 (\frac{0}{1})$

Step 3.5.2

Divide $0$ by $1$ .

$12 \cdot 0$

Step 3.5.3

Multiply $12$ by $0$ .

$0$

Step 4

List the horizontal asymptotes:

$y = 0$

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: $y = 0$

No Oblique Asymptotes

Step 7