displaystyle lim x arrow 0 (∫0x2 sin √t d t/x3) is equal to

Question

displaystyle lim x arrow 0 (∫0x2 sin √t d t/x3) is equal to (A) (2/3) (B) (1/3) (C) 0 (D) ∞

Tardigrade Admin · Accepted Answer

displaystyle lim x arrow 0 (∫ limits0x2 sin √t d t/x3) ( text form (0/0)) = displaystyle lim x arrow 0 ( sin √x2 × 2 x/3 x2) (using L' Hospital's rule) = displaystyle lim x arrow 0 (2/3) ( sin x/x)=(2/3) × 1=(2/3)

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

$\frac{2}{3}$

$\frac{1}{3}$

$0$

$\infty$

$x \to 0 lim \frac{0 \int x ^{2} sin t d t}{x ^{3}}$
$(form \frac{0}{0})$
$= x \to 0 lim \frac{sin x ^{2} \times 2 x}{3 x ^{2}}$
(using L' Hospital's rule)
$= x \to 0 lim \frac{2}{3} \frac{sin x}{x} = \frac{2}{3} \times 1 = \frac{2}{3}$

Q. x→0lim​x3∫0x2​sint​dt​ is equal to

32​

31​

0

∞

x→0lim​x30∫x2​sint​dt​ ( form 00​) =x→0lim​3x2sinx2​×2x​ (using L' Hospital's rule) =x→0lim​32​xsinx​=32​×1=32​

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

$\frac{2}{3}$

$\frac{1}{3}$

$0$

$\infty$

$x \to 0 lim \frac{0 \int x ^{2} sin t d t}{x ^{3}}$
$(form \frac{0}{0})$
$= x \to 0 lim \frac{sin x ^{2} \times 2 x}{3 x ^{2}}$
(using L' Hospital's rule)
$= x \to 0 lim \frac{2}{3} \frac{sin x}{x} = \frac{2}{3} \times 1 = \frac{2}{3}$