Limits

$\lim _{x \rightarrow \infty}\left(\frac{x^{2}}{3 x-2}-\frac{x}{3}\right)=$

If $f(x)= \begin{cases}\frac{\sin [x]}{[x]} & \text { for }[x] \neq 0 \\ 0 & \text { for }[x]=0\end{cases}$ where, $[x]$ denotes the greatest integer less than or equal to $x$, then $\lim _{x \rightarrow 0} f(x)$ is…

Q345 (image question)

$\lim _{x \rightarrow 0}\left(1^{\operatorname{cosec}^2 x}+2^{\operatorname{cosec}^2 x}+\ldots+n^{\operatorname{cosec}^2 x}\right)^{\sin ^2 x}$ is equal to

$\lim _{x \rightarrow 0} \frac{\sin 3 x-\sin x}{\sin x}$ is

$\lim _{x \rightarrow 0} \frac{e^{\sin x}-1}{x}$ is equal to

$\lim _{x \rightarrow 0} \frac{1-\cos m x}{1-\cos n x}$ is equal to

$\lim _{x \rightarrow 0}\left(\frac{(2+x) \sin (2+x)-2 \sin 2}{x}\right)$ is equal to

$\lim _{x \rightarrow 2} \frac{2-\sqrt{2+x}}{2^{1 / 3}-(4-x)^{1 / 3}}$ is equal to

$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\left(1-\tan \frac{x}{2}\right)(1-\sin x)}{\left(1+\tan \frac{x}{2}\right)(\pi-2 x)^3}$ is equal to

The value of $\lim _{\theta \rightarrow 0} \frac{\tan \theta}{\theta}$ is

If $f(x)=\frac{\sin \left(e^{x-2}-1\right)}{\log (x-1)}$, then $\lim _{x \rightarrow 2} f(x)$ is given by