JEE Main 2025 (Online) 2nd April Evening ShiftQuestion 1
If $\lim_{x\to0}\frac{\cos(2x)+a\cos(4x)-b}{x^{4}}$ is finite, then $(a+b)$ is equal to :
Limits & Continuity
Solution
Concept. For the limit over $x^4$ to be finite, the constant and $x^2$ terms of the numerator must be $0$. Formulas used. (1) $\cos u=1-\frac{u^2}{2}+\frac{u^4}{24}-\dots$Step 1 - Expand. By (1): $\cos2x=1-2x^2+\dots$, $\cos4x=1-8x^2+\dots$ Step 2 - Constant term. $1+a-b=0$. Step 3 - $x^2$ term. $-2-8a=0\Rightarrow a=-\frac14$. Then $b=1+a=\frac34$. Step 4. $a+b=-\frac14+\frac34=\frac12$. Answer: A