$ \lim_{x \rightarrow \infty} \frac{(\int_{0}^{x}e^{t^2}dt)^2}{\int_{0}^{x}e^{2t^2}dt} $ is equal to

Q: $\large \lim_{x \rightarrow \infty} \frac{(\int_{0}^{x}e^{t^2}dt)^2}{\int_{0}^{x}e^{2t^2}dt} $ is equal to

(A) 1

(B) 0

(C) –1

(D) none of these

Sol: Given limit $\large = \lim_{x \rightarrow \infty} \frac{(\int_{0}^{x}e^{t^2}dt)^2}{\int_{0}^{x}e^{2t^2}dt} $

$\large = \lim_{x \rightarrow \infty} \frac{2 \int_{0}^{x}e^{t^2}dt .e^{x^2}}{e^{2x^2}} $

$\large = \lim_{x \rightarrow \infty} \frac{2 \int_{0}^{x}e^{t^2}dt }{e^{x^2}} $

$\large = \lim_{x \rightarrow \infty} \frac{2 e^{x^2}}{e^{x^2}. 2x } = 0 $

Hence (B) is the correct answer.