Limits

$\lim _{x \rightarrow \infty}\left(\frac{x^{2}}{3 x-2}-\frac{x}{3}\right)=$

$\lim _\limits{x \rightarrow 0} \dfrac{a^x-b^x}{c^x-d^x}=$

$\lim _{a \rightarrow \pi / 2} \frac{\frac{\pi}{2}-\theta}{\cot \theta}=$

$\lim _{x \rightarrow \infty}\left(\sqrt{a^{2} x^{2}+b x+x}-a x\right)=$

$\lim _{x \rightarrow 0}\left(\frac{1+5 x^{2}}{1+3 x^{2}}\right)^{\frac{1}{x^{2}}}=$

$\lim _{n \rightarrow \infty} \frac{\left(1^{2}+2^{2}+\ldots+n^{2}\right) \sqrt[n]{n}}{(n+1)(n+10)(n+100)}=$

$\lim _{x \rightarrow \dfrac{\pi}{4}} \dfrac{\sec ^2 x-2}{\tan x-1}=$

$\lim _{x \rightarrow 0} \dfrac{\sqrt{1-\cos 2 x}}{x \sqrt{2}}$

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…

What is $\lim\limits_{x\to1}\dfrac{x^{(n^2-1)}-1}{x^{(n+1)}-1}$ equal to, where $n>1$ is a natural number ?

What is $\lim _{x \rightarrow 0} \frac{\cos x}{\pi-x}$ equal to?

For the following three (03) items: Consider the function $f(x) = x|x|$. What is $\lim_{x \to -1} f(x)$ equal to?