PUNE MCA Exam 2015 Question Paper with answers for online practice

Concept:Analyze continuity and differentiability of the function $f(x) = e^{-|x|}$.Explanation:• For $x \ge 0$, $f(x) = e^{-x}$; for $x < 0$, $f(x) = e^{x}$.• At $x=0$: left-hand limit $\lim_{x\to 0^-} e^{x} = 1$, right-hand limit $\lim_{x\to 0^+} e^{-x} = 1$, and $f(0)=1$, so continuous at $0$.• For any other $x$, $f$ is a composition of continuous functions, hence continuous everywhere.• Differentiability at $x=0$: left derivative $f'_-(0) = e^{0} = 1$, right derivative $f'_+(0) = -e^{-0} = -1$. Since they are unequal, $f$ is not differentiable at $x=0$.Answer:The function is continuous everywhere but not differentiable at $x=0$.