Limits

$$ \operatorname{Lim}_{n \rightarrow \infty} \frac{1^p+2^p+3^p+\ldots+n^p}{n^{p+1}} $$ is

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 $\underset{x\rightarrow\,0}{\lim\,}\frac{ae^{x}-b\cos\,x+ce^{-x}}{x\sin\,x}=2$, then $a+b+c$ is equal to ________.

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

Let $f,g$ and $h$ be the real valued functions defined on$\mathbb{R}$ as $f\left(x\right)=\left{\begin{matrix} \frac{x}{\left|x\right|}, & x\neq\,0 \\ 1, & x=0 \end{matrix}\right,g\left(x\right)=\left{\begin{matrix}…

Let$f\left(x\right)=\left{\begin{matrix} x-1,x\text{is even}, \\ 2x,x\text{is odd}, \end{matrix}\rightx\in\,N$. If for some$a\in\,N,f\left(f\left(f\left(a\right)\right)\right)=21$,…

If $\alpha\,=\underset{x\rightarrow\,\pi\,/4}{\lim\,}\frac{\tan\,^{3}x-\tan\,x}{\cos\,\left(x+\frac{\pi\,}{4}\right)}$ and $\beta\,=\underset{x\rightarrow\,0}{\lim\,}\left(\cos\,x\right)^{\cot\,x}$ are the roots of the…

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

If $\underset{x\rightarrow\,0}{\lim\,}\frac{ax-\left(e^{4x}-1\right)}{ax\left(e^{4x}-1\right)}$ exists and is equal to $b$, then the value of $a-2b$ is ___ .

Let $f\left(x\right)$ be a differentiable function at $x=a$ with $f^{'}\left(a\right)=2$ and $f\left(a\right)=4$. Then $\underset{x\rightarrow\,a}{\lim\,}\frac{xf\left(a\right)-af\left(x\right)}{x-a}$equals:

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

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