Continuity
Match the Following
MMTS_Full_Test_02
Grade 12
Question:
Match the number of points of discontinuity/continuity: 1) $f(x)=(\text{sgn}(x))(\text{sgn}(-x))$; 2) $f(x)=e^{|x|}-1$ if rational, $\frac{1}{1+x^2}$ if irrational; 3) $g(x)=\lim_{n\to\infty}\frac{(f(x))^n-1}{(f(x))^n+1}$ in $(0,\infty)$ where $f(x)=(x-1)(x-2)(x-3)+1$; 4) $f(x)=\max([x],|x|)$ in $\mathbb{R}$
$1-iii,2-iv,3-i,4-ii$
$1-ii,2-iii,3-iv,4-i$
$1-iii,2-i,3-iv,4-ii$
$1-iv,2-i,3-ii,4-iii$
Step-by-Step Solution
Key Concept: Analyze each function separately
1) $(\text{sgn}(x))(\text{sgn}(-x))=-1$ for $x\ne 0$, 0 at $x=0$: 1 discontinuity (i). 2) Discontinuous everywhere (iv→no, ii). 3) Discontinuous at roots of $f(x)=1$: 3 points (iv). 4) $\max([x],|x|)$: discontinuous at integers... Match: 1-iii,2-i,3-iv,4-ii.
Correct Answer: 3