Limits

$\underset{x\rightarrow\,\frac{\pi\,}{4}}{lim}\frac{cot^{3}x-tanx}{cos\left(x+\frac{\pi\,}{4}\right)}$ is

$\lim _{x \rightarrow 0}\left(\frac{x-\sin x}{x}\right) \sin \left(\frac{1}{x}\right)$

If $\lim _{x \rightarrow 0} \frac{\mathrm{e}^{(\mathrm{a}-1) x}+2 \cos \mathrm{~b} x+(\mathrm{c}-2) \mathrm{e}^{-x}}{x \cos x-\log _{\mathrm{e}}(1+x)}=2$, then $\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}$ is equal to :

Let$a$ be an integer such that $\underset{x\rightarrow\,7}{\lim\,}\frac{18-\left[1-x\right]}{\left[x-3a\right]}$exists, where$\left[t\right]$ is greatest integer $\leq\,t$. Then $a$is equal to

If $\lim _{x \rightarrow 0} \frac{\log (3+x)-\log (3-x)}{x}=k$, the value of $k$ is

If $\underset{x\rightarrow\,\infty\,}{\lim\,}\left(\sqrt{x^{2}-x+1}-ax\right)=b$, then the ordered pair $(a,b)$ is:

Let $f(x)=\begin{cases} x-1, & x\text{ is even}\\ 2x, & x\text{ is odd}\end{cases},\ x\in\mathbb{N}$. If for some $a\in\mathbb{N}$, $f(f(f(a)))=21$, then $\lim_{x\to…

Let$f:\rightarrow\,R\rightarrow\,\left(0,\infty\right)$ be strictly increasing function such that$\underset{x\rightarrow\,\infty}{\lim\,}\frac{f\left(7x\right)}{f\left(x\right)}=1$. Then, the value…

Let$\left{x\right}$denote the fractional part of$x$and$f\left(x\right)=\frac{\cos\,^{-1}\left(1-\left{x\right}^{2}\right)\sin\,^{-1}\left(1-\left{x\right}\right)}{\left{x\right}-\left{x\right}^{3}},x\neq\,0$.If…

If $\operatorname{Lim}_{x \rightarrow 0}\left(\frac{\tan x}{x}\right)^{\frac{1}{x^2}}=p$, then $96 \log _e p$ is equal to ______

If $\alpha=\lim _{x \rightarrow 0^{+}}\left(\frac{\mathrm{e}^{\sqrt{\tan x}}-\mathrm{e}^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}\right)$ and $\beta=\lim _{x \rightarrow 0}(1+\sin x)^{\frac{1}{2} \cot x}$ are the roots of the…

$\lim _{x \rightarrow 0} \frac{\sin \left(\pi \cos ^2 x\right)}{x^2}$ equals