Question

Given $f(x)=\{\begin{array}{ll}3-\left[{\cot }^{-1}\left(\frac{2x^3-3}{x^2}\right)\right]& \text{ for }x>0\\ \left\{x^2\right\}\cos \left(e^{1/x}\right)& \text{ for }x<0\end{array}$ 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)

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