Continuity and Differentiability
Continuity at x=0 — Piecewise Absolute Value
nta_pyq_2026_jan
Grade 12
Question:
If $f(x) = \begin{cases} \dfrac{a|x|+x^2-2\sin|x|\cos|x|}{x} & x\neq0 \\ b & x=0 \end{cases}$ is continuous at $x=0$, then $a+b$ is equal to
Step-by-Step Solution
Key Concept: Note $2\sin|x|\cos|x|=\sin 2|x|$. For $x>0$: $f(x)=a+x-\dfrac{\sin 2x}{x}\xrightarrow{x\to0^+} a-2$. For $x<0$: $f(x)=-a+x+\dfrac{\sin 2x}{x}\xrightarrow{x\to0^-}-a+2$.
$a=2$, $b=0$. $a+b=2$.
Correct Answer: 3