Question

$f\left(x\right)=\{\begin{array}{cc}\frac{\tan \left[x\right]-\left[x\right]\tan 1}{x}& ;x\ne 0\\ 0& ;x=0\end{array}$, then $f^{\prime }\left(0^{-}\right)$ is $[$where $\left[x\right]$ denotes integer part of $x]$

• A. $0$
• B. $1$
• C. $-1$
• D. Does not exist

Answer 1

The correct option is A $0$

$\because f\left(0\right)=0$
$\Rightarrow f^{\prime }\left(0^{-}\right)=\underset{h\to 0}{lim}\frac{f(-h)-f\left(0\right)}{-h},h>0$

$=\underset{h\to 0}{lim}\frac{\tan [-h]-[-h]\tan 1}{h^2}$

$=\frac{-\tan 1+\tan 1}{0^{+}}=0$