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Chapter 2: Problem 52

Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$g(x)=e^{1 / x}$$

Short Answer

Expert verified
Answer: The vertical asymptote of the function $$g(x) = e^{1/x}$$ is at $$x = 0$$.

Step by step solution

01

Identify the function and domain

The given function is $$g(x) = e^{1/x}$$. The function is defined for all real numbers, except $$x = 0$$, as the expression becomes undefined when the denominator is 0.
02

Analyze the limit as x approaches the undefined value

Since the function becomes undefined at $$x = 0$$, we need to check the behavior of the function as $$x$$ approaches 0 from both the left (negative) side and the right (positive) side. For $$x \rightarrow 0^{-}$$: $$\lim_{x \rightarrow 0^{-}} e^{1/x}$$ For $$x \rightarrow 0^{+}$$: $$\lim_{x \rightarrow 0^{+}} e^{1/x}$$
03

Evaluate the limits

As we know the function $$e^{x}$$ increases as $$x$$ increases and decrease as $$x$$ decreases. Also, as $$x$$ approaches 0, the exponential term approaches either positive or negative infinity. For $$x \rightarrow 0^{-}$$: $$\lim_{x \rightarrow 0^{-}} e^{1/x} = e^{-\infty} = 0$$ For $$x \rightarrow 0^{+}$$: $$\lim_{x \rightarrow 0^{+}} e^{1/x} = e^{\infty} = \infty$$
04

Identify the vertical asymptote(s)

Since the limits, as x approaches 0 from both the left and right side, are not equal, we can conclude that there is a vertical asymptote at x = 0. Therefore, the vertical asymptote of the given function is: $$x = 0$$

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Most popular questions from this chapter

Analyzing infinite limits graphically Graph the function \(y=\tan x\) with the window \([-\pi, \pi] \times[-10,10] .\) Use the graph to analyze the following limits. a. \(\lim _{x \rightarrow \pi / 2^{+}} \tan x\) b. \(\lim _{x \rightarrow \pi / 2^{-}} \tan x\) d. \(\quad \lim _{n}\) tan \(x\) c. \(\lim _{x \rightarrow-\pi / 2^{+}} \tan x\) d. \(\lim _{x \rightarrow-\pi / 2^{-}} \tan x\)

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