Calculus Examples

$y = \frac{3}{e^{x} - 2}$

Step 1

Find where the expression $\frac{3}{e^{x} - 2}$ is undefined.

$x = \ln (2)$

Step 2

Evaluate $lim_{x \to \infty} \frac{3}{e^{x} - 2}$ to find the horizontal asymptote.

Tap for more steps...

Step 2.1

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

$3 lim_{x \to \infty} \frac{1}{e^{x} - 2}$

Step 2.2

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

$3 \cdot 0$

Step 2.3

Multiply $3$ by $0$ .

$0$

Step 3

Evaluate $lim_{x \to - \infty} \frac{3}{e^{x} - 2}$ 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 $3$ outside of the limit because it is constant with respect to $x$ .

$3 lim_{x \to - \infty} \frac{1}{e^{x} - 2}$

Step 3.1.2

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

$3 \frac{lim_{x \to - \infty} 1}{lim_{x \to - \infty} e^{x} - 2}$

Step 3.1.3

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

$3 \frac{1}{lim_{x \to - \infty} e^{x} - 2}$

Step 3.1.4

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

$3 \frac{1}{lim_{x \to - \infty} e^{x} - lim_{x \to - \infty} 2}$

Step 3.2

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

$3 \frac{1}{0 - lim_{x \to - \infty} 2}$

Step 3.3

Evaluate the limit.

Tap for more steps...

Step 3.3.1

Evaluate the limit of $2$ which is constant as $x$ approaches $- \infty$ .

$3 \frac{1}{0 - 1 \cdot 2}$

Step 3.3.2

Simplify the answer.

Tap for more steps...

Step 3.3.2.1

Simplify the denominator.

Tap for more steps...

Step 3.3.2.1.1

Multiply $- 1$ by $2$ .

$3 \frac{1}{0 - 2}$

Step 3.3.2.1.2

Subtract $2$ from $0$ .

$3 (\frac{1}{- 2})$

Step 3.3.2.2

Move the negative in front of the fraction.

$3 (- \frac{1}{2})$

Step 3.3.2.3

Multiply $3 (- \frac{1}{2})$ .

Tap for more steps...

Step 3.3.2.3.1

Multiply $- 1$ by $3$ .

$- 3 (\frac{1}{2})$

Step 3.3.2.3.2

Combine $- 3$ and $\frac{1}{2}$ .

$\frac{- 3}{2}$

Step 3.3.2.4

Move the negative in front of the fraction.

$- \frac{3}{2}$

Step 4

List the horizontal asymptotes:

$y = 0, - \frac{3}{2}$

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: $x = \ln (2)$

Horizontal Asymptotes: $y = 0, - \frac{3}{2}$

No Oblique Asymptotes

Step 7