Question

Examine whether the given function f(x) is continuous at x=3 .

$f\left(x\right)=\begin{array}{c}\frac{x^4-8x}{\sqrt{x^2+5}-3},forx\ne 3\\ 3forx=3\end{array}$
$f\left(x\right)=3$

Answer 1

Given $f\left(x\right)=\begin{array}{cc}\frac{x^4-8x}{\sqrt{x^2+5}-3}& forx\ne 3\\ =3,& forx=3\end{array}$

$\underset{x\to 3}{lim}\frac{x^4-8x}{\sqrt{x^2+5}-3}$

$=\frac{81-24}{\sqrt{14}-3}=\frac{57}{\sqrt{14}-3}$

$\therefore \underset{x+3}{lim}l\left(n\right)\ne l\left(3\right)$

Hence, $l\left(n\right)$ is not continuous at $x=3$