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  • Evaluate: <img alt="\mathrm{: \lim _{n \rightarrow \infty}\left(\frac{3^{-n} \sin \left(3^{(1-n)}\right)}{\tan \left(3^{1-2 n}\right)}\right)}" src="https://entrancecorner.oncodecogs.com/gif.latex?

Evaluate: \mathrm{: \lim _{n \rightarrow \infty}\left(\frac{3^{-n} \sin \left(3^{(1-n)}\right)}{\tan \left(3^{1-2 n}\right)}\right)}

Option: 1

0


Option: 2

4


Option: 3

1


Option: 4

2


Answers (1)

best_answer

\mathrm{\text { Putting } h=3^{-n}}

\mathrm{\text { As } n \rightarrow \infty \Longrightarrow h \rightarrow 0}

\mathrm{\text { and } 3^{1-n}=3 \cdot 3^{-n}=3 h ; 3^{1-2 n}=3 \cdot\left(3^{-n}\right)^2=3 h^2}

                                                                     \mathrm{\lim _{n \rightarrow \infty}\left(\frac{3^n \sin \left(3^{(1-n)}\right)}{\tan \left(3^{1-2 n}\right)}\right)}

                                                                    \mathrm{=\lim _{h \rightarrow 0}\left(\frac{h \sin (3 h)}{\tan \left(3 h^2\right)}\right)}

                           \mathrm{=\left(\lim _{h \rightarrow 0} \frac{\sin 3 h}{3 h}\right) \cdot\left(\lim _{h \rightarrow 0} \frac{3 h^2}{\sin 3 h^2}\right) \cdot\left(\lim _{h \rightarrow 0} \cos \left(3 h^2\right)\right)}

Posted by

Pankaj Sanodiya

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