Let f : R $$ \to $$ R be twice continuously differentiable (or f" exists and is continuous) such that f(0) = f(1) = f'(0) = 0. Then

Let $$0 < \alpha < \beta < 1$$. Then, $$\mathop {\lim }\limits_{n \to \infty } \int\limits_{1/(k + \beta )}^{1/(k + \alpha )} {{{dx} \over {1 + x}}} $$ is

$$\mathop {\lim }\limits_{x \to 1} \left( {{1 \over {1nx}} - {1 \over {(x - 1)}}} \right)$$

$$\mathop {\lim }\limits_{x \to {0^ + }} ({x^n}\ln x),\,n > 0$$