Question

Let $[.]$ denote the greatest integer function and $f\left(x\right)=\left[{\tan }^{2}x\right]$.Then

• A. $\underset{x\to 0}{lim}f\left(x\right)$does not exist
• B. $f\left(x\right)$ is continuous at $x=0$
• C. $f\left(x\right)$ is not differentiable at $x=0$
• D. $f\left(0\right)=1$

Answer 1

The correct option is B $f\left(x\right)$ is continuous at $x=0$

${lim}_{x\to 0^{-}}\left(ta{n}^{2}x\right)=0$
${lim}_{x\to 0^{+}}\left(ta{n}^{2}x\right)=0$
L.H.S.=R.H.S. so${lim}_{x\to 0}\left(ta{n}^{2}x\right)=0$
& it is continous at $x=0$
$f\left(x\right)=\left(ta{n}^{2}x\right)$
or $f\left(x\right)=c$ Where c $ϵ$ Integers
so $f^1\left(x\right)=0$
$f\left(x\right)isdifferentiableatx=0$