Science:Math Exam Resources/Courses/MATH110/April 2019/Question 01 (c)

MATH110 April 2019

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Question 01 (c)
Let $f(x)={\begin{cases}2x^{2}-3&{\text{if }}x<1\\{\frac {1}{3x}}&{\text{if }}x\geq 1.\end{cases}}$ Is $f$ continuous at $x=1$ ? Explain why.

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Given a function $f$ , and a point $x_{0}$ in the domain of $f$ , what does it mean for the function $f$ to be continuous at the point $x_{0}$ ? This will involve checking left- and right-hand limits, but should include more information than this.

Hint 2
In order to determine whether the function $f$ given in the question is continuous at $x=1$ , you will need to compare the following quantities: $\lim _{x\rightarrow 1^{+}}f(x)$ $\lim _{x\rightarrow 1^{-}}f(x)$ $f(1)$ itself, no limits here. In order for $f$ to be continuous, what must be true about these three quantities, and how must they be related to one another?

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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 are asked to determine whether the function $f$ defined in the problem statement is continuous at the point $x=1$ We'll see that the function is discontinuous at $x=1$ . The easiest way to see this is to compare the left- and right-hand limits of the function $f$ at the point $x=1$ . A simple calculation gives: $\lim _{x\to 1^{-}}f(x)=\lim _{x\rightarrow 1^{-}}2x^{2}-3=-1$ and $\lim _{x\to 1^{+}}f(x)=\lim _{x\rightarrow 1^{+}}{\frac {1}{3x}}=1/3$ . Therefore, the left- and right-hand limits are not equal. Examining the definition of continuity, we see that $f$ must be discontinuous at $x=1$ , since these one-sided limits are not equal.