લક્ષ અને વિકલન — ગણિત ધોરણ 11 સાયન્સ — Question

$\lim_{x \rightarrow \infty} \frac{\int_{0}^{2x} xe^2dx }{e^{4x^2}}$

$=\lim_{x \rightarrow \infty} \frac{\int_{0}^{2x}xe^{x^2}dx}{e^{4x^2}}$

$=\lim_{x \rightarrow \infty} \frac{\int_{0}^{2x}e^{x^2}dx}{2e^{4x^2}}$

$=\lim_{x \rightarrow \infty} \frac{[e^{x^2}]_{0}^{2x}}{2e^{4x^2}}$

$=\lim_{x \rightarrow \infty} \frac{[e^{x^2}]^{2x}_0}{2e^{4x^2}} \Rightarrow \lim_{x \rightarrow \infty} \frac{e^{4x^2}-1}{2e^{4x^2}}$

$=\lim_{x \rightarrow \infty} \left(\frac{1}{2}- \frac{1}{2e^{4x^2}}\right)$

$=\frac{1}{2}-\frac{1}{\infty}$

$=\frac{1}{2}-0$

$=\frac{1}{2}$