KVPY Sample Paper KVPY Stream-SX Model Paper-5

Let \[f(x):\frac{\tan \,[{{e}^{2}}]{{x}^{3}}-\tan \,[-\,{{e}^{2}}]{{x}^{3}}}{{{\sin }^{3}}x},x\ne 0\] ([.] represents greatest integer function). The value of \[f(0)\] for which \[f(x)\] is continuous, is

A) 15

B) 12

C) \[-\,12\]

D) 14

Correct Answer: A

Solution :

[a]

We have, \[f(x)=\frac{\tan [{{e}^{2}}]{{x}^{3}}-\tan [-{{e}^{2}}]{{x}^{3}}}{{{\sin }^{3}}x}\]

\[7<{{e}^{2}}<8,\] so \[[{{e}^{2}}]=7\] and \[[-\,{{e}^{2}}]=-\,8\]

So \[f(0)=\underset{x\to 0}{\mathop{\lim }}\,\frac{\tan 7{{x}^{3}}-\tan (-\,8){{x}^{3}}}{{{\sin }^{3}}x}\]

\[\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{\tan 7{{x}^{3}}}{{{\sin }^{3}}x}+\frac{\tan 8{{x}^{3}}}{{{\sin }^{3}}x} \right]\]\[=7+8=15\]