Question

$\lim_{x \rightarrow 0}$$\int_{0}^{x^{2}}{\sin\sqrt{t}}$ dt is equal to –

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

Answer 1

Answer: (B)

$\frac{2}{3}$

Answer 2

$\lim _ { x \rightarrow 0 }$ $\int _ { 0 } ^ { \mathrm { x } ^ { 2 } } \sin \sqrt { \mathrm { t } }$ dt

=$\lim _ { x \rightarrow 0 }$ $\frac { \int _ { 0 } ^ { x ^ { 2 } } \sin \sqrt { t } d t } { x ^ { 3 } }$ $\left( \frac { 0 } { 0 } \right.$ form $)$ $\left( \because \int _ { 0 } ^ { \mathrm { x } ^ { 2 } } \sin \sqrt { \mathrm { t } } \mathrm { dt } = 0 \right.$ at $\left. \mathrm { x } = 0 \right)$

= $\lim _ { x \rightarrow 0 }$ . (1)

=$\lim _ { x \rightarrow 0 }$ = $\frac { 2 } { 3 }$

Hence (2) is the correct answer.