Calculus Examples

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

Step 1

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

$x = 0$

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

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

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Step 3.1

Evaluate the limit.

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Step 3.1.1

Move the limit into the exponent.

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

Step 3.1.2

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 3.2

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

$e^{- 0}$

Step 3.3

Simplify the answer.

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Step 3.3.1

Multiply $- 1$ by $0$ .

$e^{0}$

Step 3.3.2

Anything raised to $0$ is $1$ .

$1$

Step 4

List the horizontal asymptotes:

$y = 1$

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: $y = 1$

No Oblique Asymptotes

Step 7