First slide

Introduction to limits

Question

$lim_{x \to \infty} \frac{\int_{0}^{2 x} x e^{x^{2}} d x}{e^{4 x^{2}}}$ equals

Moderate

A

0
B

$\infty$
C

2
D

$\frac{1}{2}$

Solution

We have,
$lim_{x \to \infty} \frac{\int_{0}^{2 x} x e^{x^{2}} d x}{e^{4 x^{2}}} = lim_{x \to \infty} \frac{\int_{0}^{2 x} e^{x^{2}} d (x^{2})}{2 e^{4 x^{2}}} = lim_{x \to \infty} \frac{{[e^{x^{2}}]}_{0}^{2 x}}{2 e^{4 x^{2}}}$
$\Rightarrow lim_{x \to \infty} \frac{\int_{0}^{2 x} x e^{x^{2}} d x}{e^{4 x^{2}}} = lim_{x \to \infty} \frac{e^{4 x^{2}} - 1}{2 e^{4 x^{2}}} = lim_{x \to \infty} (\frac{1}{2} - \frac{1}{e^{4 x^{2}}}) = \frac{1}{2}$

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