Graph f(x)=e^(1/x)

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

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

$x = 0$

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

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

Move the limit into the exponent.

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

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

$e^{0}$

Anything raised to $0$ is $1$ .

$1$

List the horizontal asymptotes:

$y = 1$

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

This is the set of all asymptotes.

No Vertical Asymptotes

Horizontal Asymptotes: $y = 1$

No Oblique Asymptotes