List of top Mathematics Questions on Limits asked in JEE Advanced

Evaluate the limit: \[ \lim_{x \to 0} \frac{\frac{\alpha}{2} \int_0^x \frac{dt}{1 - t^2} + \beta x \cos x}{x^3} = \gamma \]

Let \(k\in R\). If \(\)\(\)\(\)\(\lim\limits_{x→0+}(\sin(\sin kx)+\cos x+x)^{\frac{2}{x}}=e^6\), then the value of k is

Let S be the set of all \((α, β) ∈ \R\times\R\) such that
\(\lim\limits_{x\rightarrow\infin}\frac{\sin(x^2)(\log_ex)^\alpha\sin(\frac{1}{x^2})}{x^{αβ}(\log_e(1+x))^β}=0\)
Then which of the following is (are) correct ?

For positive integer $n$, define $f(n)=n+\frac{16+5 n-3 n^2}{4 n+3 n^2}+\frac{32+n-3 n^2}{8 n+3 n^2}+\frac{48-3 n-3 n^2}{12 n+3 n^2}+\ldots+\frac{25 n-7 n^2}{7 n^2}$.
Then, the value of $\displaystyle\lim _{n \rightarrow \infty} f(n)$ is equal to

The value of the limit
$\displaystyle\lim _{x \rightarrow \frac{\pi}{2}} \frac{4 \sqrt{2}(\sin 3 x+\sin x)}{\left(2 \sin 2 x \sin \frac{3 x}{2}+\cos \frac{5 x}{2}\right)-\left(\sqrt{2}+\sqrt{2} \cos 2 x+\cos \frac{3 x}{2}\right)}$
is _______

For non-negative integers n, let $f\left(n\right) = \frac{\displaystyle\sum^{n}_{k = 1}sin\left(\frac{k+1}{n+2}\pi\right) sin \left(\frac{k+2}{n+2}\pi\right)}{\displaystyle\sum^{n}_{k = 1} sin^{2}\left(\frac{k+1}{n+2}\pi\right)}$ Assuming $cos^{-1}x$ takes values in $\left[0, \pi\right]$, which of the following options is/are correct?

For x > 0, $ lim_{ x \to 0} \Bigg [ (sin \, x)^{1/x} + \bigg( \frac{1}{x}\bigg)^{sin \, x} \Bigg ] $ is

If $ lim_{x \to 0 } \frac{ \{(a - n) nx - tan \, x \} sin \, nx }{x^2} = 0, $ where n is non-zero real number, then a is equal to

$ lim_{ h \to 0 } \frac{ f (2 h + 2 + h^2 ) - f \, (2)}{ f \, ( h - h^2 + 1) - f (1)} $, given thta f ' (2) = 6 and f ' (1) = 4.

$ lim_{ n \to \infty} \frac{1}{n} \displaystyle \sum_{r = 1}^ {2n} \frac{r}{ \sqrt{ n^2 + r^2}} $ equals