Verified Solution • 2025–26 Session moderate

QJEEM25Ma5c91: JEE Main 2025 Mathematics - Class 12 JEEM25Ma5c91 Solution: Let Then is equal to ___________.

Let f(x)=lim⁡n→∞∑r=0n(tan⁡(x/2r+1)+tan⁡3(x/2r+1)1−tan⁡2(x/2r+1))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⁡x→0ex−ef(x)(x−f(x))\lim\limits_{x \rightarrow 0} \frac{e^x-e^{f(x)}}{(x-f(x))} is equal to ___________.

Step-by-Step Verified Solution

f(x)=limn→∞∑r=0n(tan⁡x2r−tan⁡x2r+1)=tan⁡xlimx→0(ex−etan⁡xx−tan⁡x)=limx→0etan⁡x(ex−tan⁡x−1)(x−tan⁡x)=1

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