Calculus Examples

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

Step 1

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

$x = 0$

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

Move the limit into the exponent.

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

Step 3.2

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

$\frac{1}{1 + e^{0}}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Anything raised to $0$ is $1$ .

$\frac{1}{1 + 1}$

Step 3.3.2

Add $1$ and $1$ .

$\frac{1}{2}$

Step 4

List the horizontal asymptotes:

$y = \frac{1}{2}$

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.

No Vertical Asymptotes

Horizontal Asymptotes: $y = \frac{1}{2}$

No Oblique Asymptotes

Step 7