Question

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

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

Answer 1

Answer: (B)

$f\left(x\right)$ is continuous at $x=0$

$f\left(x\right)=\left[{\tan }^{2}x\right]$

at $x=0$

as $x\to 0^{-},\tan x>0\Rightarrow {\tan }^{2}x>0$

$x\to 0^{+},\tan x<0\Rightarrow {\tan }^{2}x>0$

${lim}_{x\to 0^{-}}\left[{\tan }^{2}x\right]=0$

${lim}_{x\to 0^{+}}\left[{\tan }^{2}x\right]=0$

LHLRHL$=$RHL $\Rightarrow$continuous at $x=0$