Limits JEE Main PYQs, 2020 to 2025

Given below are two statements:

Statement I: $\lim\limits_{x \to 0} \left( \frac{\tan^{-1} x + \log_e \sqrt{\frac{1+x}{1-x}} - 2x}{x^5} \right) = \frac{2}{5}$

Statement II: $\lim\limits_{x \to 1} \left( x^{\frac{2}{1-x}} \right) = \frac{1}{e^2}$

In the light of the above statements, choose the correct answer from the options given below:

$\lim \limits_{x \rightarrow 0^{+}} \frac{\tan \left(5(x)^{\frac{1}{3}}\right) \log _e\left(1+3 x^2\right)}{\left(\tan ^{-1} 3 \sqrt{x}\right)^2\left(e^{5(x)^{\frac{4}{3}}}-1\right)}$ is equal to

Let [t] be the greatest integer less than or equal to t. Then the least value of p ∈ N for which

$\lim\limits_{x \to 0^+} \left( x (\left[ \frac{1}{x} \right] + \left[ \frac{2}{x} \right] + \ldots + \left[ \frac{p}{x} \right] \right) - x^2 \left( \left[ \frac{1}{x^2} \right] + \left[ \frac{2^2}{x^2} \right] + \ldots + \left[ \frac{9^2}{x^2} \right] \right) \geq 1$ is equal to _______.

$$\lim \limits_{x \rightarrow \frac{\pi}{2}}\left(\frac{\int_{x^3}^{(\pi / 2)^3}\left(\sin \left(2 t^{1 / 3}\right)+\cos \left(t^{1 / 3}\right)\right) d t}{\left(x-\frac{\pi}{2}\right)^2}\right)$$ is equal to

Let $f: \rightarrow \mathbb{R} \rightarrow(0, \infty)$ be strictly increasing function such that $\lim \limits_{x \rightarrow \infty} \frac{f(7 x)}{f(x)}=1$. Then, the value of $\lim \limits_{x \rightarrow \infty}\left[\frac{f(5 x)}{f(x)}-1\right]$ is equal to