Question

Check the continuity of the function.
$f\left(x\right)=\{\begin{array}{c}\frac{x^4+x^3+2x^2}{{\tan }^{-1}x}ifx\ne 0\\ 10ifx=0\end{array}$

Answer 1

$\begin{array}{c}f\left(x\right)willbesurelycontinuousatallpoints\\ butwehavetocheckitscontinuityatx=0\\ forf\left(x\right)tobecontinuousatx=0\\ {lim}_{x\to 0}f\left(x\right)=f\left(0\right)\\ \Rightarrow {lim}_{x\to 0}\frac{x^4+x^3+2x^2}{{\tan }^{-1}x}=f\left(0\right)\\ usingLhospitalrule\\ \Rightarrow {lim}_{x\to 0}\frac{4x^3+3x^2+4x}{\frac{1}{1+x^2}}=f\left(0\right)\\ =0\ne f\left(0\right)\\ hencef\left(x\right)iscontinuousatallpointsexceptx=0\end{array}$