If $0 < x < y$,then $\mathop {\lim }\limits_{n \to \infty } {({y^n} + {x^n})^{1/n}}$ is equal to

The value of $\mathop {\lim }\limits_{n \to \infty } \frac{{1 + 2 + 3 + .... + n}}{{{n^2} + 100}}$ is equal to:

$\mathop {\lim }\limits_{x \to \infty } \sqrt {\frac{{x + \sin x}}{{x - \cos x}}} = $

$\lim _{\theta \rightarrow \frac{\pi}{2}^{-}} \frac{8 \tan ^4 \theta+4 \tan ^2 \theta+5}{(3-2 \tan \theta)^4} = $

For each $t \in \mathbb{R}$,let $[t]$ be the greatest integer less than or equal to $t$. Then $\lim_{x \to 1^+} \frac{(1 - |x| + \sin |1 - x|) \sin (\frac{\pi}{2} [1 - x])}{|1 - x|^2}$ is:

$\lim _{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 a}}{\sqrt{x}-\sqrt{a}} = $