Introduction to Limits

If $\underset{x\to \infty }{\lim }\frac{\left(e^{\mu x}+5\right)}{\left(e^{100x}+7\right)}$ exists, then sum of all possible positive integral values of $\mu$ is

It is given that $\underset{x\to 0}{\lim }\frac{\frac{x^2}{{\left(a+x^r\right)}^{{\displaystyle \frac{1}{p}}}}}{1-\cos \mathit{x}}=\ell$. If $P=3$ and $\ell =1$, then the value of $a$ is equal to

Evaluate the $\underset{n\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{1}{n^3}\sum _{k=1}^n\left[k^{2}x\right]$, (where $\left[x\right]$ denotes greatest integer function less than or equal to $x$).

Define physical interpretation of $\frac{dy}{dx}$ and find $\frac{dy}{dx}$ for $y=8x^5$.

Define physical interpretation of $\frac{dy}{dx}$ and find $\frac{dy}{dx}$ for $y=3x^2+2x$.

Define physical interpretation of $\frac{dy}{dx}$ and find $\frac{dy}{dx}$ for $y=12x^4$.

Define physical interpretation of $\frac{dy}{dx}$ and find $\frac{dy}{dx}$ for $y=5x^3+4x$.

Define physical interpretation of $\frac{dy}{dx}$ and find $\frac{dy}{dx}$ for $y=4x^3$.

Find a function $y=g\left(x\right)$ when the gradient function is $\frac{dy}{dx}=\frac{x}{2}$.

Find a function $y=g\left(x\right)$ when the gradient function is $\frac{dy}{dx}=-2$.

Find a function $y=g\left(x\right)$ when the gradient function is $\frac{dy}{dx}=3$.

If $f\left(a\right)=a^2,g\left(a\right)=b^2$ and $f\text{'}\left(a\right)=kg\text{'}\left(a\right),$ then show that $\underset{x\to a}{\lim }\frac{\sqrt{f\left(x\right)}-a}{\sqrt{g\left(x\right)}-b}=\frac{kb}{a.}$ 

If $f\left(4\right)=4, f\text{'}\left(4\right)=8,$ then find the value of
$\underset{x\to 4}{\lim }\frac{\sqrt{f\left(x\right)}-2}{\sqrt{x}-2}.$

Evaluate $\underset{x\to 1}{\lim }\frac{x^2-\sqrt{x}}{\sqrt{x}-1}$.

Evaluate $\underset{x\to 3}{\lim }\frac{x^2-9}{x-3}$

Evaluate:$\underset{x\to 1}{\lim }\frac{x^2+x+1}{x^2+2x+100}$.

If $\underset{\text{x}\to 0^{+}}{\text{lim}}\text{f}\left(\text{x}\right)=$ finite, where $\text{f}\left(\text{x}\right)=\frac{\text{sin x}+{\text{ae}}^{\text{x}}+{\text{be}}^{-\text{x}}+\text{c ln}\left(1+\text{x}\right)}{{\text{x}}^3}$ and $\mathrm{a},\mathrm{b},\mathrm{c}$ are real numbers.

Then the value of $\mathrm{b}$ is 

The value of $f\left(0\right)$, so that the function $f\left(x\right)=\frac{1-\cos \left(1-\cos x\right)}{x^4}$ is continuous everywhere is $k$, then the value of $10k$ is

If $\underset{x\to 1}{\lim }\left(\frac{1}{x-1}-\frac{2}{x^2-1}\right)=\frac{m}{n}$, then find the value of $n-m$.

If $\underset{x\to \infty }{\lim }\left(\frac{x^2+1}{x+1}-ax-b\right)=0$, then