Science:Math Exam Resources/Courses/MATH110/December 2017/Question 03 (c)

MATH110 December 2017

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Question 03 (c)
Evaluate the following limits or determine they are either infinite or do not exist. (c) ${\textstyle \displaystyle \lim _{x\to 0}{\frac {x^{2}-1}{x^{2}+x^{3}}}}$

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?

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Hint
What are the limits of the numerator and denominator, respectively?

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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. Let ${\textstyle f(x)=x^{2}-1}$ and ${\textstyle g(x)=x^{2}+x^{3}}$. Note that ${\textstyle f(x)}$ and ${\textstyle g(x)}$ are polynomials and hence continuous everywhere. Following the Hint, we compute $$\lim _{x\to 0}f(x)=0^{2}-1=-1$$ and $$\lim _{x\to 0}g(x)=0^{2}+0^{3}=0.$$ For ${\textstyle x}$ close enough to ${\textstyle 0}$, we get ${\textstyle -1}$ divided by a very small number. Therefore, $$\lim _{x\to 0}{\dfrac {f(x)}{g(x)}}=-\infty .$$ Notice that the negative sign comes from the ${\textstyle f(x)}$. The sign of ${\textstyle g(x)=x^{2}(1+x)}$ is positive near $0$ because ${\textstyle 1+x}$ is close to ${\textstyle 1}$. Answer: ${\textstyle \lim \limits _{x\to 0}{\dfrac {x^{2}-1}{x^{2}+x^{3}}}={\color {blue}-\infty }.}$