If $f(x) = \begin{cases} x, & \text{when } 0 \le x \le 1 \\ 2 - x, & \text{when } 1 < x \le 2 \end{cases}$,then $\lim_{x \to 1} f(x) = $

If ${S_n} = \sum\limits_{k = 1}^n {{a_k}} $ and $\mathop {\lim }\limits_{n \to \infty } {a_n} = a,$ then $\mathop {\lim }\limits_{n \to \infty } \frac{{{S_{n + 1}} - {S_n}}}{{\sqrt {\sum\limits_{k = 1}^n k } }}$ is equal to

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

For the function $f(x) = (1 + \frac{1}{x})^x$,which of the following limits tends to unity?

$\mathop {\lim }\limits_{x \to 1} \frac{{1 - {x^{ - 1/3}}}}{{1 - {x^{ - 2/3}}}} = $

If $\mathop {\lim }\limits_{x \to \infty } {\left( {1 + \frac{a}{x} + \frac{b}{{{x^2}}}} \right)^{2x}} = {e^2},$ then the values of $a$ and $b$ are