Methods of Evaluation of Limits

Let $a_1>a_2>a_3>\dots >a_n>1; p_1>p_2>p_3>\dots >p_n>0;$ such that $p_1+p_2+p_3+\dots +p_n=1$. Also $F\left(x\right)={\left(p_1{a}_1^x+p_2{a}_2^x+\dots +p_n{a}_n^x\right)}^{1/x},$ then $\underset{x\to \infty }{\lim }F\left(x\right)$ equals

If $\underset{x\to 0}{\lim }\frac{1-cosx.cos2x.cos3x\dots \dots \dots \cos nx}{x^2}$ has the value equal to$253$, find the value of $n$ is equal to (where $n\in N$)

The value of $\underset{x\to 0}{\lim }\frac{x^{6000}-{\left(\sin x\right)}^{6000}}{x^2{\left(\sin x\right)}^{6000}}$ is

The value of $\underset{x\to 0}{\lim }\frac{1+sinx-cosx+\ln \left(1-x\right)}{x·{tan}^{2}x}$ is

$\underset{x\to 0}{\lim }\frac{x+\ln \left(\sqrt{1+x^2}-x\right)}{x^3}=$

$\underset{x\to 0}{\lim }\frac{8}{x^8}\left[1-\cos \frac{x^2}{2}-\cos \frac{x^2}{4}+\cos \frac{x^2}{2}\cos \frac{x^2}{4}\right]$

Let n be an odd integer, if  $\sin n\theta ={\displaystyle \sum _{r=0}^n{b}_r{\sin }^r\theta }$ , for every value of  $\theta$ , then

Let $\alpha$ be a positive real number. Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ and $g:\left(\alpha ,\infty \right)\to \mathrm{ℝ}$ be the functions defined by $f\left(x\right)=\sin \left(\frac{\pi x}{12}\right)$ and $g\left(x\right)=\frac{2{\log }_e\left(\sqrt{x}-\sqrt{\alpha }\right)}{{\log }_e\left(e^{\sqrt{x}}-e^{\sqrt{\alpha }}\right)}$.
Then the value of $\underset{x\to {\alpha }^{+}}{\lim }f\left(g\left(x\right)\right)$ is ______.

$\underset{y\to 0}{\lim }\frac{(y-2)+2\sqrt{1+y+y^2}}{2y}$ is equal to___________

$\underset{x\to 0}{\lim }\frac{\log \left(\sin 7x+\cos 7x\right)}{\sin 3x}$ equals

If $f\left(x\right)=\left|\begin{array}{ccc}\sin x& \cos x& \tan x\\ x^3& x^2& x\\ 2x& 1& 1\end{array}\right|$, then $\underset{x\to 0}{\lim }\frac{f\left(x\right)}{x^2}$ is

$\underset{x\to 1}{\lim }\frac{4^x-4}{\sin \left(x-1\right)}=$

Evaluate:$\underset{x\to 0}{\lim }\frac{e^{ax}-1}{\sin bx}=$

$\underset{x\to 3}{\lim }\frac{x^n-3^n}{x-3}=27n$, then $n=$

$\underset{x\to 0}{\lim }\frac{\sin x}{\sin x}=$

$\underset{x\to \infty }{\lim } \frac{2x-{\ln }^{2}x}{\sqrt{x} \ln x+x}$ is equal to

The value of $\underset{x\to 0}{\lim }\frac{{\left(\cos x\right)}^{\frac{1}{2}}-{\left(\cos x\right)}^{\frac{1}{3}}}{{\left(\cos x\right)}^{\frac{1}{2}}-{\left(\cos x\right)}^{\frac{1}{5}}}$ is equal to

If $\left[x\right]$ is the greatest integer function, then $\underset{x\to 0}{\lim } x^5\left[\frac{1}{x^4}\right]$ is equal to 

If $\underset{n\to \infty }{\lim }{\left(\frac{n^2+n+2}{n^2+2n+3}\right)}^n=\frac{p}{e^q}$, then $p+q$ is

$\underset{x\to \infty }{\lim } \frac{2x-{\ln }^{2}x}{\sqrt{x} \ln x+x}$ is equal to