Limits, Continuity & Differentiability
Continuity and Differentiability of Log-exponential Function
nta_pyq_2024_jan
Grade 12
Question:
Consider the function $f:(0,\infty)\to\mathbb{R}$ defined by $f(x)=e^{-|\log_e x|}$. If $m$ and $n$ be respectively the number of points at which $f$ is not continuous and $f$ is not differentiable, then $m+n$ is
Step-by-Step Solution
Key Concept: $f(x)=e^{-|\ln x|}=\begin{cases}e^{\ln x}=x & 0<x<1\\e^{-\ln x}=1/x & x\ge1\end{cases}$. Check continuity at $x=1$: both branches give 1. ✓ Check differentiability at $x=1$: LHD$=1$, RHD$=-1$. Not differentiable at $x=1$.
$f(x)=\begin{cases}x & 0<x<1\\1/x & x\ge1\end{cases}$.
Continuous at $x=1$: $\lim_{x\to1^-}x=1=f(1)=1$. ✓ $m=0$.
Differentiability at $x=1$: LHD$=1$, RHD$=-1$. Not differentiable. $n=1$.
$m+n=0+1=1$.
Correct Answer: 3