Q53 / 120 · VITEEE 2008 · Mathematics
Let [ ] denote the greatest integer function and $f(x)=\left[\tan ^{2} x\right]$. Then
- A$\lim _{x \rightarrow 0} f(x)$ does not exist
- Correct answerB$f(x)$ is continuous at $x=0$
- C$f(x)$ is not differentiable at $x=0$
- D$\mathrm{f}(\mathrm{x})=1$
Solution
Check the continuity of the function
$\begin{array}{l}
\mathrm{f}(\mathrm{x})=\left[\tan ^{2} \mathrm{x}\right] \text { at } \mathrm{x}=0 . \\
\text { L.H.L. (at } \mathrm{x}=0 \text { ) }
\end{array}$
$\begin{array}{l}
=\lim _{x \rightarrow 0^{-}}\left[\tan ^{2} x\right]=\lim _{h \rightarrow 0}\left[\tan ^{2}(0-h)\right] \\
=\lim _{h \rightarrow 0}\left[\tan ^{2} h\right]=\left[\tan ^{2} 0\right]=[0]=0 \\
\text { R.H.L. }(\operatorname{at} x=0) \\
=\lim _{x \rightarrow 0^{+}}\left[\tan ^{2} x\right]=\lim _{h \rightarrow 0}\left[\tan ^{2}(0+h)\right]
\end{array}$
$=\lim _{h \rightarrow 0}\left[\tan ^{2} h\right]=\left[\tan ^{2} 0\right]=[0]=0$
Now, determine the value of $f(x)$ at $x=0$. $\mathrm{f}(0)=\left[\tan ^{2} 0\right]=[0]=0$
Hence, $f(x)$ is continuous at $x=0$.
Hence, the correct answer is option: 'B'.