The limit displaystyle lim x arrow ∞ x2 ∫ limits0x et3-x3 d t equals

Question

The limit displaystyle lim x arrow ∞ x2 ∫ limits0x et3-x3 d t equals (A) (1/3) (B) 2 (C) ∞ (D) (2/3)

Tardigrade Admin · Accepted Answer

We have, displaystyle limx→∞ x2∫ limits0x et3 -x3 dt = displaystyle limx→∞ x2e-x3 ∫ limits0x et3dt = displaystyle limx→∞ (x2∫ limits0xet3dt/ex3) Apply L Hospitalâs rule = displaystyle limx→∞ (2x∫ limits0xet3dt+x2ex3/3x2ex3) = displaystyle limx→∞ (2∫ limits0xet3dt+x2ex3/3x2 ex3) Again Apply L Hospitalâs rule, we get = displaystyle limx→∞ (2ex3+ex3+3x3ex3/3ex3+9x3ex3) = displaystyle limx→∞ (ex3(3+3x3)/ex3(3+9x3))=(1/3)

Q. The limit $x \to \infty lim x^{2} 0 \int x e^{t^{3} - x^{3}} d t$ equals

$\frac{1}{3}$

$2$

$\infty$

$\frac{2}{3}$

Q. The limit x→∞lim​x20∫x​et3−x3dt equals

31​

2

∞

32​

Q. The limit $x \to \infty lim x^{2} 0 \int x e^{t^{3} - x^{3}} d t$ equals

$\frac{1}{3}$

$2$

$\infty$

$\frac{2}{3}$