Science:Math Exam Resources/Courses/MATH110/April 2019/Question 01 (c)/Solution 1

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:

$\underset{x\to 1^{-}}{\lim }f(x)=\underset{x\to 1^{-}}{\lim }2x^2-3=-1$ and $\underset{x\to 1^{+}}{\lim }f(x)=\underset{x\to 1^{+}}{\lim }\frac{1}{3x}=1/3$.

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.