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BITSAT 2015
Exam
BITSAT
Year
2015
Subject
Mathematics
Chapter
Limits

BITSAT 2015 · Mathematics · Limits

_ x arrow ∞ ∫_ 0 ^ 2 x x e^ x^ 2 d x e^ 4 x^ 2 equals

lim⁡x→∞∫02xxex2dxe4x2\lim _{x \rightarrow \infty} \frac{\int_{0}^{2 x} x e^{x^{2}} d x}{e^{4 x^{2}}} equals \text{ equals }

Options

  1. A0‾\underline{0}
  2. B∞\infty
  3. C2
  4. D12\frac{1}{2}

Correct answer

D. 12\frac{1}{2}

Step-by-step solution

Consider lim⁡x→∞∫02xxex2dxe4x2\lim _{x \rightarrow \infty} \frac{\int_{0}^{2 x} x e^{x^{2}} d x}{e^{4 x^{2}}}=lim⁡x→∞2∫02xxex2dx2e4x2=lim⁡x→∞2∫02xex2d(x2)2e4x2=lim⁡x→∞[ex2]02xe4x2=lim⁡x→∞e4x2−12e4x2=lim⁡x→∞(12−12e4x2)=12\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}$$
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