Let f(x) be a differentiable function at x = a with f'(a) = 2 and f(a) = 4.

Then $\mathop {\lim }\limits_{x \to a} {{xf(a) - af(x)} \over {x - a}}$ equals :

$$\mathop {\lim }\limits_{x \to 0} {{x\left( {{e^{\left( {\sqrt {1 + {x^2} + {x^4}} - 1} \right)/x}} - 1} \right)} \over {\sqrt {1 + {x^2}…

$\lim _\limits{x \rightarrow 0^{+}} \frac{\tan \left(5(x)^{\frac{1}{3}}\right) \log _e\left(1+3 x^2\right)}{\left(\tan ^{-1} 3…

If $\alpha$ is positive root of the equation, p(x) = x2 - x - 2 = 0, then $$\mathop {\lim }\limits_{x \to {\alpha ^ + }} {{\sqrt {1 - \cos…

$$\mathop {\lim }\limits_{n \to \infty } \tan \left\{ {\sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {{1 \over {1 + r + {r^2}}}} \right)} }…

Let $f(x) = \left[ {2{x^2} + 1} \right]$ and $$g(x) = \left\{ {\matrix{ {2x - 3,} & {x