How do you find the vertical, horizontal or slant asymptotes for # f(x) = e^(1/x)#?

Question

Elijah Allen · Accepted Answer

Horizontal asymptote: y = 1.
Vertical asymptote: x = 0..
The left #limxto0-# of y = 0.

The exponential function #e^(1/x) > 0# The whole graph is above y = 1, in the first quadrant, and below y = 1, in the third.. For graph in the first quadrant: As x increases from 0+, y decreases towards 1. For graph in the third quadrant: As x decreases from #0-#, y increases from 0 towards 1. Note that the left #limxto0-# of y = 0. This is the case of an asymptote meeting the other branch of the curve, in the opposite direction.

Benjamin Achtenberg · Answer

undefined To find the vertical asymptotes of $ f(x) = e^{1/x} $, we set the denominator of the function, $ x $, equal to zero and solve for $ x $. In this case, $ x = 0 $ is a vertical asymptote.

To determine horizontal or slant asymptotes, we examine the behavior of the function as $ x $ approaches positive or negative infinity.

As $ x $ approaches positive infinity, $ e^{1/x} $ approaches $ e^0 = 1 $. Therefore, there is a horizontal asymptote at $ y = 1 $.

As $ x $ approaches negative infinity, $ e^{1/x} $ approaches $ e^0 = 1 $. Similarly, there is a horizontal asymptote at $ y = 1 $.

Since the function does not exhibit any unbounded behavior as $ x $ approaches positive or negative infinity, there are no slant asymptotes. Therefore, the vertical asymptote is $ x = 0 $, and the horizontal asymptote is $ y = 1 $.