Question

Given $f(x)=\begin{cases}3-\left[\cot ^{-1}\left(\frac{2 x^3-3}{x^2}\right)\right] & \text { for } x>0 \\ \left\{x^2\right\} \cos \left(e^{1 / x}\right) & \text { for } \quad x<0\end{cases}$ where $\{\}$ & [ ] denotes the fractional part and the integral part functions respectively, then which of the following statement does not hold good -

• A. $f\left(0^{-}\right)=0$
• B. $f\left(0^{+}\right)=3$
• C. $f(0)=0 \Rightarrow$ continuity of $f$ at $x=0$
• D. irremovable discontinuity of $f$ at $x=0$

Answer 1

Answer: (B), (D)

$RHL =\displaystyle\lim _{x \rightarrow 0^{+}}\left(3-\left[\cot ^{-1}\left(\frac{2 x^3-3}{x^2}\right)\right]\right)$
$=3-\left[\cot ^{-1}(-\infty)\right]=3-3=0$
$LHL =\displaystyle\lim _{h \rightarrow 0}\left\{(0-h)^2\right\} \cos \left(e^{\left(\frac{1}{0-h}\right)}\right)$
$=\displaystyle\lim _{h \rightarrow 0}(0-h)^2 \cos \left(e^{-\infty}\right)=0$