If $\mathop {\lim }\limits_{x \to 0} \phi (x) = {a^3}, (a \ne 0)$; then $\mathop {\lim }\limits_{x \to 0} \phi \left( {\frac{x}{a}} \right)$ is equal to :-

$\lim _{x \rightarrow 0} \frac{e^x-e^{\sin x}}{2(x-\sin x)}$

The value of $\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{{x^2} + bx + 4}}{{{x^2} + ax + 5}}} \right)$ is

Let $a_1, a_2, a_3, \ldots, a_n$ be $n$ positive consecutive terms of an arithmetic progression. If $d > 0$ is its common difference,then evaluate $\lim_{n \rightarrow \infty} \sqrt{\frac{d}{n}} \left( \frac{1}{\sqrt{a_1} + \sqrt{a_2}} + \frac{1}{\sqrt{a_2} + \sqrt{a_3}} + \ldots + \frac{1}{\sqrt{a_{n-1}} + \sqrt{a_n}} \right)$.

The value of $\lim _{x \rightarrow 1} \frac{x+x^2+\ldots+x^n-n}{x-1}$ is

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) = $