displaystyle lim x arrow 0 (∫ limits0x2( sin √t) d t/x3) is equal to

Question

displaystyle lim x arrow 0 (∫ limits0x2( sin √t) d t/x3) is equal to (A) (2/3) (B) (3/2) (C) 0 (D) (1/15)

Tardigrade Admin · Accepted Answer

displaystyle lim x arrow 0+ (∫ limits0x2 sin √t d t/x3)= displaystyle lim x arrow 0+ (( sin x) 2 x/3 x2) = displaystyle lim x arrow 0+(( sin x/x)) × (2/3)=(2/3)

Q. $x \to 0 lim \frac{0 \int x ^{2} ( sin t ) d t}{x ^{3}}$ is equal to

$\frac{2}{3}$

$\frac{3}{2}$

0

$\frac{1}{15}$

$x \to 0^{+} lim \frac{0 \int x ^{2} sin t d t}{x ^{3}} = x \to 0^{+} lim \frac{( sin x ) 2 x}{3 x ^{2}}$
$= x \to 0^{+} lim (\frac{sin x}{x}) \times \frac{2}{3} = \frac{2}{3}$

Q. x→0lim​x30∫x2​(sint​)dt​ is equal to

32​

23​

0

151​

x→0+lim​x30∫x2​sint​dt​=x→0+lim​3x2(sinx)2x​ =x→0+lim​(xsinx​)×32​=32​

Q. $x \to 0 lim \frac{0 \int x ^{2} ( sin t ) d t}{x ^{3}}$ is equal to

$\frac{2}{3}$

$\frac{3}{2}$

$\frac{1}{15}$

$x \to 0^{+} lim \frac{0 \int x ^{2} sin t d t}{x ^{3}} = x \to 0^{+} lim \frac{( sin x ) 2 x}{3 x ^{2}}$
$= x \to 0^{+} lim (\frac{sin x}{x}) \times \frac{2}{3} = \frac{2}{3}$