Science:Math Exam Resources/Courses/MATH110/December 2011/Question 01 (c)

MATH110 December 2011

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Question 01 (c)
Evaluate the following limit. $\lim _{x\to \infty }\left(e^{-x}-x\right)$

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
What does the graph of $e^{-x}$ look like? What happens to it as x goes to infinity?

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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. We will consider the function inside the limit as the sum of two functions: $e^{-x}$ and $-x$. If we take their limits separately, we get: $\displaystyle \lim _{x\to \infty }e^{-x}=0$ $\displaystyle \lim _{x\to \infty }-x$ diverges to $\displaystyle -\infty$ We can see that as x goes to infinity, the $e^{-x}$ term is decreasing towards zero and the $-x$ term grows larger and larger. So, as x goes to infinity the combined value of $e^{-x}-x$ will grow more and more negative. Thus we have: $\displaystyle \lim _{x\to \infty }e^{-x}-x$ diverges to $\displaystyle -\infty$