Limits

Q4513 (image question)

Q4514 (image question)

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

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…

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:

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

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 ________.

The value of $\underset{n\rightarrow\,\infty\,}{\lim\,}\frac{\left[r\right]+\left[2r\right]+...+\left[nr\right]}{n^{2}},$where $r$is non-zero real number and $\left[r\right]$ denotes the greatest integer less than or…

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

If$\alpha\,$ is the positive root of the equation,$p\left(x\right)=x^{2}-x-2=0$,then $\underset{x\rightarrow\,\alpha\,^{+}}{\lim\,}\frac{\sqrt{1-\cos\,p\left(x\right)}}{x+\alpha\,-4}$is equal 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 $x=2$be a root of the equation $x^{2}+px+q=0$and$f\left(x\right)=\left{\begin{matrix} \frac{1-\cos\,\left(x^{2}-4px+q^{2}+8q+16\right)}{\left(x-2p\right)^{4}}, & x\neq\,2p \\ 0, & x=2p \end{matrix}\right$.Then…