Question

If $[\cdot]$ denotes the greatest integer function, then $\displaystyle\lim _{x \rightarrow 0} \frac{\tan \left(\left[-2 \pi^{2}\right] x^{2}\right)-x^{2} \tan \left[-2 \pi^{2}\right]}{\sin ^{2} x}$ is equal to

• A. $-20+\tan 20$
• B. $20+\tan 20$
• C. $20$
• D. None of these

Answer 1

Answer: (A)

We have, $\left[-2 \pi^{2}\right]=-20$
$\Rightarrow \displaystyle\lim _{x \rightarrow 0} \frac{\tan \left(\left[-2 \pi^{2}\right] x^{2}\right)-x^{2} \tan \left[-2 \pi^{2}\right]}{\sin ^{2} x}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{\tan \left(-20 x^{2}\right)-x^{2} \tan (-20)}{\sin ^{2} x}$
$=\displaystyle\lim _{x \rightarrow 0}-\frac{\tan 20 x^{2}}{20 x^{2}} \times 20 \times \frac{x^{2}}{\sin ^{2} x}+\left(\frac{x}{\sin x}\right)^{2} \tan 20$
$=-20+\tan 20$