Science:Math Exam Resources/Courses/MATH110/April 2018/Question 07 (a)

MATH110 April 2018

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Question 07 (a)
Let $f(x)={\frac {e^{x}}{x}}.$ (a) Find all vertical asymptotes, if they exist. Justify your answer.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
The vertical line $x=a$ is a vertical asymptote of a function $f$ if one of the one–sided limits as $x\to a^{-}$ or $x\to a^{+}$ of $f$ is infinite.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. By the quotient rule of limits, if the limit of the denominator $\lim _{x\to a}x=a\neq 0$, then we have $$\lim _{x\to a}f(x)=\lim _{x\to a}{\frac {e^{x}}{x}}={\frac {\lim _{x\to a}e^{x}}{\lim _{x\to a}x}}={\frac {e^{a}}{a}}.$$ Since for any $a\neq 0$, ${\frac {e^{a}}{a}}$ has a finite value, $x=0$ is the only candidate of a vertical asymptote of $f$. To see if $x=0$ is indeed a vertical asymptote, we compute the one-side limit. When $x$ is close to $0$ and positive, $e^{x}$ is close to $e^{0}=1$ and ${\frac {1}{x}}$ is a very very large positive number. It follows that $$\lim _{x\to 0^{+}}f(x)=\lim _{x\to 0+}{\frac {e^{x}}{x}}=\lim _{x\to 0+}e^{x}\cdot {\frac {1}{x}}=+\infty .$$ Therefore, by the definition of vertical asymptote in the Hint, $x=0$ is the only vertical asymptote of the function $f$. Answer: $\color {blue}x=0$