AIEEE 2003 (Offline)Question 1
The value of $\lim_{x\to0}\frac{\int_{0}^{x^{2}}\sec^{2}t\,dt}{x\sin x}$ is
Limits & Continuity
Solution
Concept. The integral can be done exactly: $\int\sec^2t\,dt=\tan t$. Formulas used. (1) $\int_0^{x^2}\sec^2t\,dt=\tan x^2$(2) $\lim_{u\to0}\frac{\tan u}{u}=1$, $\ \lim_{x\to0}\frac{\sin x}{x}=1$Step 1 - Integrate (1). Expression $=\frac{\tan x^2}{x\sin x}$. Step 2 - Split (2). $$\frac{\tan x^2}{x^2}\cdot\frac{x}{\sin x}\to1\cdot1=1$$ Answer: D