Question

Given $f\left(x\right)=\{\begin{array}{cc}3-\left[{\cot }^{-1}\left(\frac{2x^3-3}{x^2}\right)\right]& \text{for}x>0\\ \left\{x^2\right\}\cos \left(e^{1/x}\right)& \text{for}x<0\end{array}$ 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. $f\left(0\right)=0\Rightarrow continuityoffatx=0$
• D. irremovable discontinuity of f at $x=0$

Answer 1

Answer: (A), (B), (D)

$f\left(0^{+}\right)=3$
$f\left(0^{-}\right)=0$
irremovable discontinuity of f at $x=0$

RHL $=\underset{x\to 0^{+}}{lim}(3-[{\cot }^{-1}\left(\frac{2x^3-3}{x^2}\right)\left]\right)$
$=3-\left[{\cot }^{-1}\right(-\infty \left)\right]=3-0=3$
LHL $=\underset{h\to 0}{lim}\left\{\right(0-h{)}^2\}\cos \left(e^{\left(\frac{1}{0-h}\right)}\right)$
$=\underset{h\to 0}{lim}(0-h{)}^2\cos (e^{-\infty })=0$
Clearly discontinuity is irremovable, (since L.H.L$\ne$ R.H.L)