Science:Math Exam Resources/Courses/MATH110/December 2014/Question 03 (b)

MATH110 December 2014

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Question 03 (b)
Which one of the following statements is the best reason why ${\textstyle f(x)}$ is not continuous at ${\textstyle x=0}$? (i) ${\textstyle f(0)}$ does not exist. (ii) ${\textstyle \displaystyle \lim _{x\to 0}f(x)}$ does not exist. (iii) Both (i) and (ii). (iv) ${\textstyle f(0)}$ and ${\textstyle \displaystyle \lim _{x\to 0}f(x)}$ exist, but they are not equal.

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 that $f$ is continuous at $x=a$ if the following are all satisfied; $f$ is defined at $x=a$ $\lim _{x\to a}f(x)$ exists $f(a)=\lim _{x\to a}f(x)$

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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. From the graph of $f$ in part (a), we can easily see that $f(0)=1$, but $\lim _{x\to 0-}f(x)=1\neq 0=\lim _{x\to 0+}f(x)$. Therefore, the given function is not continuous, because the limit $\lim _{x\to 0}f(x)$ doesn't exist. Answer: $\color {blue}(ii)$