Science:Math Exam Resources/Courses/MATH110/December 2016/Question 04 (b)

MATH110 December 2016

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Question 04 (b)
Determine whether each one of the following functions is continuous at $x=1$ using the definition of continuity. You must fully justify your answer. (b) ${\begin{aligned}f(x)={\begin{cases}{\frac {x^{2}-1}{x-1}}&{\text{if }}x\neq 1\\2&{\text{if }}x=1\end{cases}}\end{aligned}}$

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
Recall the definition of continuity given in the Hint of Question 4(a).

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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 observe that ${\textstyle f(1)=2}$ is defined. Now we compute the left hand limit and the right hand limit. In fact, since the formula is the same on both sides, we simply need to compute $$\lim _{x\to 1}f(x)=\lim _{x\to 1}{\dfrac {x^{2}-1}{x-1}}=\lim _{x\to 1}{\dfrac {(x-1)(x+1)}{x-1}}=\lim _{x\to 1}(x+1)=1+1=2.$$ In other words, the left hand and right hand limits both exist and are equal to ${\textstyle 2}$, which is the value of ${\textstyle f(1)}$. Therefore, ${\textstyle f(x)}$ is continuous at ${\textstyle x=1}$. Answer: ${\textstyle f(x)}$ is ${\textstyle {\color {blue}{\mbox{continuous}}}}$ at ${\textstyle x=1}$.