Find the limit: $\mathop {\lim }\limits_{x \to 1} \frac{x^{2}+1}{x+100}$

$\lim _{n \rightarrow \infty} \frac{n(2 n+1)^2}{(n+2)(n^2+3 n-1)}$

Let $f(x)=5-|x-2|$ and $g(x)=|x+1|$,$x \in R$. If $f(x)$ attains its maximum value at $\alpha$ and $g(x)$ attains its minimum value at $\beta$,then $\lim _{x \rightarrow-\alpha \beta} \frac{(x-1)(x^2-5x+6)}{(x^2-6x+8)}$ is equal to

$\lim_{x \rightarrow \frac{\pi}{2}} (\tan^{2} x (\sqrt{2 \sin^{2} x + 3 \sin x + 4} - \sqrt{\sin^{2} x + 6 \sin x + 2}))$ is equal to

Find the limit: $\mathop {\lim }\limits_{x \to 3} [x(x+1)]$

$\mathop {\lim }\limits_{x \to a} \frac{{{{(x + 2)}^{5/3}} - {{(a + 2)}^{5/3}}}}{{x - a}} = $