Q32 / 90 · KVPY 2016 (SB/SX) · Mathematics

Limit $\lim _{x \rightarrow \infty} x^{2} \int_{0}^{x} e^{t^{3}-x^{3}}$ dt equals

  • Correct answer
    A
    $\frac{1}{3}$
  • B
    2
  • C
    $\infty$
  • D
    $\frac{2}{3}$

Solution

$$ \lim _{x \rightarrow \infty} \frac{x^{2} \int_{0}^{x} e^{x^{3}} d t}{e^{x^{3}}} $$ Apply L Hospital $$ \begin{array}{l} \lim _{x \rightarrow \infty} \frac{2 x \int_{0}^{x} e^{t^{3}}+x^{2} e^{x^{3}}}{3 x^{2} e^{x^{3}}} \\ \lim _{x \rightarrow \infty} \frac{2 \int_{0}^{x} e^{t^{3}}+x e^{x^{3}}}{3 x e^{x^{3}}} \\ \lim _{x \rightarrow \infty} \frac{2 e^{x^{3}}+e^{x^{3}}+3 x^{3} e^{x^{3}}}{3 e^{x^{3}}+9 x^{3} e^{x^{3}}} \\ \lim _{x \rightarrow \infty} \frac{3+3 x^{3}}{3+9 x^{3}}=\frac{1}{3} \end{array} $$

Hence, the correct answer is option: 'A'.

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