Question

Find if the given below function is continuous or discontinuous at the indicated point
$f\left(x\right)=\{\begin{array}{ccc}x+5,& \text{if}& x\ge 2\\ x^2,& \text{if}& x<2\end{array}$ at x = 2

Answer 1

Find right hand limit at $x=2$

Given that,
$f\left(x\right)=\{\begin{array}{ccc}x+5,& \text{if}& x\ge 2\\ x^2,& \text{if}& x<2\end{array}$

Find right hand limit at $x=2$
$R.H.L=\underset{x\to 2^{+}}{\text{lim}}f\left(x\right)$
$=\underset{x\to 2^{+}}{\text{lim}}f(2+h)$
$\underset{h\to 0^{+}}{\text{lim}}\left(3\right(2+h)+5)=11$ ...(1)

Find left hand limit at $x=2$

$LHL=\underset{x\to 2^{-}}{\text{lim}}f\left(x\right)$
$=\underset{h\to 0^{+}}{\text{lim}}f(2-h)=\underset{h\to 0^{+}}{\text{lim}}(2-h{)}^2=4$ ...(2)

Since $LHL\ne RHL$ at $x=2$

limit does not exist.

Thus, $f\left(x\right)$ is discontinuous at $x=2$