JEE MAIN - Mathematics (2025 - 28th January Evening Shift - No. 21)
Let $f(x)=\lim \limits_{n \rightarrow \infty} \sum\limits_{r=0}^n\left(\frac{\tan \left(x / 2^{r+1}\right)+\tan ^3\left(x / 2^{r+1}\right)}{1-\tan ^2\left(x / 2^{r+1}\right)}\right)$ Then $\lim\limits_{x \rightarrow 0} \frac{e^x-e^{f(x)}}{(x-f(x))}$ is equal to ___________.
Answer
1
Explanation
$$\begin{aligned}
& f(x)=\lim _{n \rightarrow \infty} \sum_{r=0}^n\left(\tan \frac{x}{2^r}-\tan \frac{x}{2^{r+1}}\right)=\tan x \\
& \lim _{x \rightarrow 0}\left(\frac{e^x-e^{\tan x}}{x-\tan x}\right)=\lim _{x \rightarrow 0} e^{\tan x} \frac{\left(e^{x-\tan x}-1\right)}{(x-\tan x)} \\
& =1
\end{aligned}$$
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