Q41 / 125 · BITSAT 2015 · Mathematics

$$\lim _{x \rightarrow \infty} \frac{\int_{0}^{2 x} x e^{x^{2}} d x}{e^{4 x^{2}}} \text { equals }$$

  • A
    $\underline{0}$
  • B
    $\infty$
  • C
    2
  • Correct answer
    D
    $\frac{1}{2}$

Solution

Consider $\lim _{x \rightarrow \infty} \frac{\int_{0}^{2 x} x e^{x^{2}} d x}{e^{4 x^{2}}}$ $$ \begin{array}{l} =\lim _{x \rightarrow \infty} \frac{2 \int_{0}^{2 x} x e^{x^{2}} d x}{2 e^{4 x^{2}}} \\ =\lim _{x \rightarrow \infty} \frac{2 \int_{0}^{2 x} e^{x^{2}} d\left(x^{2}\right)}{2 e^{4 x^{2}}} \\ =\lim _{x \rightarrow \infty} \frac{\left[e^{x^{2}}\right]_{0}^{2 x}}{e^{4 x^{2}}}=\lim _{x \rightarrow \infty} \frac{e^{4 x^{2}}-1}{2 e^{4 x^{2}}} \\ =\lim _{x \rightarrow \infty}\left(\frac{1}{2}-\frac{1}{2 e^{4 x^{2}}}\right)=\frac{1}{2} \end{array} $$

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

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