Science:Math Exam Resources/Courses/MATH110/April 2016/Question 04 (d)

MATH110 April 2016

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Question 04 (d)
Evaluate the following limits, if they exist. If a limit does not exist, specify whether it is (positive or negative) infinity or neither. (d) $\lim _{x\to 5^{-}}{\frac {x}{(x-5)^{2}}}.$

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
Note that as $x\to 5^{-}$, the bottom tends to zero while the top tends to $5^{-}$. So then which infinity is it? Positive or negative?

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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. Following the hint, as $x\to 5^{-}$, the denominator $(x-5)^{2}\to 0^{+}$ (note that the direction we approach 0 doesn't actually matter here since the denominator is squared!), while the numerator $x\to 5,$ so $\lim _{x\to 5^{-}}{\frac {x}{(x-5)^{2}}}=\infty .$ Thus, the limit does not exist, and instead goes to $\color {blue}+\infty .$