Introduction to Limits

Find the limit.

$\underset{x\to 2}{\lim }\left(\frac{\sqrt{1-\cos \left\{2\left(x-2\right)\right\}}}{x-2}\right)$

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

What is limits?

What is $\underset{x\to 5}{\lim }\frac{5-x}{\left|x-5\right|}$ equal to?

$\underset{n\to \infty }{\lim } \frac{5^{n+1}+3^n-2^{2n}}{5^n+2^n+3^{2n+3}}$ is equal to $?$

Evaluate $\underset{x\to 0}{\lim } \frac{\sqrt{7}-\sqrt{6+\cos x}}{x\sin x}$

Evaluate the limit $\underset{x\to 0}{\lim }\left(\frac{2x}{\left|x\right|+x^2}\right)$.

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

Find  $\underset{x\to 0}{\lim } \frac{e^{{\displaystyle \frac{1}{x}}}}{e^{{\displaystyle \frac{1}{x}}}+1}$

$\underset{x\to 0}{\lim }\log \left(1+5x\right)=$

$\underset{x\to 1}{\lim }\left[x\right]+\sqrt{\left\{x\right\}}$

Evaluate:

$\underset{x\to 0}{\lim }\frac{\sqrt{1+x}-1}{x}$

Find the limits:

$\underset{x\to -1}{\lim }\left[1+x+x^2+\dots +x^{10}\right]$

Find the limits:

$\underset{x\to 3}{\lim }\left[x\left(x+1\right)\right]$

Evaluate:

$\underset{x\to 2^{+}}{\lim }\frac{\left[x-2\right]}{\log \left(x-2\right)}$

$\underset{\mathrm{x}\to \frac{\mathrm{\pi }}{6}}{lim}\frac{2-\sqrt{3}\cos \mathrm{x}+\sin \mathrm{x}}{(6\mathrm{x}-\mathrm{\pi }{)}^2}$

Find the value of $\underset{x\to \frac{\mathrm{\pi }}{4}}{\lim } \frac{1-\sin 2x}{1+\cos 2x}$

Evaluate $\underset{x\to 0}{\lim } \frac{3x+\left|x\right|}{7x-5\left|x\right|}$

Let $f\left(x\right)=\left\{\begin{array}{l}\sin x; x\in Z\\ 0; \mathrm{otherwise}\end{array}\right.$ and $$\begin{gathered} g\left(x\right)=\left\{\begin{array}{l}{x}^2+1; x\ne 0,2 \\ 4; x=0\\ 5; x=2\end{array}\right. \end{gathered}$$, then

If $f\left(x\right)=\left\{\begin{array}{cc}x+2,& x<1\\ 4x-1,& 1\le x\le 3\\ x^2+5,& x>3\end{array}\right.$then find the value of $\underset{x\to 1^{+}}{\lim }f\left(x\right)$, $\underset{x\to 1^{-}}{\lim }f\left(x\right)$, $\underset{x\to 3^{+}}{\lim }f\left(x\right)$ and $\underset{x\to 3}{\lim }f\left(x\right)$