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Let [.] denote the greatest integer function and $f(x)=[\tan^2x]$ then

$\begin{array}{1 1}(a)\;\lim\limits_{x\to 0}f(x)\;does\;not\;exist\\(b)\;f(x)\;is\;continuous\;at\;x=0\\(c)\;f(x)\;is\;not\;differentiable\;at\;x=0\\(d)\;f'(0)=1\end{array}$

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We have $f(x)=[\tan^2x]$
$\tan x$ is an increasing function.
For $-\large\frac{\pi}{4}$$ < x < \large\frac{\pi}{4}$
$\Rightarrow -1 < \tan x < 1$
$\Rightarrow 0 < \tan^2 x < 1$
$\Rightarrow [\tan^2x]=0$
Hence $\lim\limits_{x\to 0}f(x)=\lim\limits_{x\to 0}[\tan^2x]=0$
$f(0)=0$
$\therefore f(x)$ is continuous at $x=0$
answered Dec 31, 2013 by sreemathi.v
 

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