A) 0
B) 1
C) \[\tan \,\,10-10\]
D) \[\infty \]
Correct Answer: C
Solution :
\[\because \] \[\pi =3,\,\,14\] \[{{\pi }^{2}}=9.86\] \[[-{{\pi }^{2}}]=[-9.86]=-10\] Then, \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\tan \,(-10{{x}^{2}})\,-{{x}^{2}}\,\tan \,(-10)}{{{\sin }^{2}}x}\] \[\Rightarrow \] \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\frac{-\tan \,10\,{{x}^{2}}}{10{{x}^{2}}}\times \frac{10{{x}^{2}}}{{{x}^{2}}}+\frac{{{x}^{2}}\,\tan \,10}{{{x}^{2}}}}{\frac{{{\sin }^{2}}x}{{{x}^{2}}}}\] \[\left( \begin{align} & \because \,\underset{\theta \to 0}{\mathop{\lim }}\,\,\frac{\sin \theta }{\theta }=1 \\ & \underset{\theta \to 0}{\mathop{\lim }}\,\,\frac{\tan \,\theta }{\theta }=1 \\ \end{align} \right)\] \[=\frac{-10+\tan \,10}{1}\] \[=\tan 10-10\]
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