Question

Discuss the continuity of the following function at x = 0 :

\(f(x) = \begin{cases} \frac{x^4+2x^3+x^2}{tan^{-1}x}&, \quad &x≠0\\ 0&, &x=0 \end{cases}\)

Answer 1

\(\lim\limits_{x \to 0^-} f(x) = \)\(\lim\limits_{h \to 0} f(0-h)\)

f(0) = 0

[ ∵ f(x) = 0 for x = 0]

i.e., 

\(\lim\limits_{x \to 0^-} f(x) = \)\(\lim\limits_{x \to 0^+} f(x) =f(0)=0\)

Hence, f(x) is continuous at x = 0.