JEE Main 2026 — 23 January, Evening Shift — Question 1

$f(x)=\left\{\begin{array}{cc}\frac{a|x|+x^{2}-2 \sin |x| \cos |x|}{x} & ; x \neq 0 \\b & x=0\end{array}\right.$

for continuity $\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)=f(0)$

$\lim _{\mathrm{x} \rightarrow 0^{-}} \frac{\mathrm{ah}+\mathrm{h}^{2}-2(\sinh ) \cosh }{-\mathrm{h}}$

$=\lim _{\mathrm{x} \rightarrow 0^{+}} \frac{\mathrm{ah}+\mathrm{h}^{2}-2(\sinh ) \cosh }{\mathrm{h}}$ or $-\mathrm{a}+2=\mathrm{a}-2=\mathrm{b}$

$2 \mathrm{a}=4$

$\mathrm{a}=2, \mathrm{~b}=0$

$\therefore \mathrm{a}+\mathrm{b}=2$