Browse Questions

# Discuss the continuity of the following function at x = 0 $f(x)= \left\{ \begin{array}{1 1} \Large\frac{x^4+x^3+2x^2}{\tan ^{-1}x} & \quad ,\;x\neq0 \\ 10 & \quad ,\;x=0 \end{array} \right.$

Toolbox:
• For continuity at '0' LHL = RHL = f(0)
• $\lim\limits_{x \to 0} \: \large\frac{tan^{-1}x}{x}=1$
LHL = RHL
$= \lim\limits_{x \to 0} \: \large\frac{\large x^4+x^3+2x^2}{tan^{-1}x}$
$= \lim\limits_{x \to 0} \: \large\frac{\large x^3+x^2+2x}{\large\frac{tan^{-1}x}{x}} = 0$

$f(0) = 10$
$\Rightarrow$ f is not continuous.

edited Mar 26, 2013