Question

$\underset{x\to 0}{\mathrm{Lim}}\frac{\underset{0}{\overset{x}{\int }}\sin t^{2}dt}{x(1-\cos x)}$ equals

• A. $\frac{1}{3}$
• B. 2
• C. $\frac{1}{2}$
• D. $\frac{2}{3}$

Answer 1

Answer: (D)

$\underset{x\to 0}{\mathrm{Lim}}\frac{\underset{0}{\overset{x}{\int }}\sin t^{2}dt}{x^3\frac{(1-\cos x)}{x^2}}$ (using $\underset{x\to 0}{\mathrm{Lim}}\frac{1-\cos x}{x^2}=\frac{1}{2}$ )
$=2\underset{x\to 0}{\mathrm{Lim}}\frac{\underset{0}{\overset{x}{\int }}\sin t^{2}dt}{x^3}$ (Using L'Hospital Rule)
$=2\underset{x\to 0}{\mathrm{Lim}}\frac{\sin x^2}{3x^2}=\frac{2}{3}$