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  • Without using the method of L' Hospital, find the limit. <img alt="\lim _{\theta \rightarrow \infty}\left(\theta \log _{e}\left(1+\frac{3^{3}}{\theta}\right)\right)" src="https://entrancec

Without using the method of L' Hospital, find the limit. \lim _{\theta \rightarrow \infty}\left(\theta \log _{e}\left(1+\frac{3^{3}}{\theta}\right)\right)

Option: 1

9


Option: 2

\frac{\ln 2}{9}+1


Option: 3

27


Option: 4

3


Answers (1)

best_answer

Note the following exponential limit.

\lim _{x \rightarrow0}\frac{\log _{e}(1+x)}{x}=1

Assume   \frac{1}{\theta}=x

So, as   \theta \rightarrow \infty, x \rightarrow 0

Now, evaluate the following limit using the above formula and the assumption.

\begin{aligned} & \lim _{\theta \rightarrow \infty}\left(\theta \log _{e}\left(1+\frac{3^{3}}{\theta}\right)\right) \\ & =\lim _{x \rightarrow 0}\left(\frac{1}{x} \log _{e}\left(1+3^{3} x\right)\right) \\ & =\lim _{x \rightarrow 0} \frac{\log _{e}(1+27 x)}{x} \\ \end{aligned}\begin{aligned} & =27 \times \lim _{x \rightarrow 0} \frac{\log _{e}(1+27 x)}{27 x} \\ & =27 \times 1 \\ & =27 \end{aligned}

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vinayak

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