I need to evaluate the following limit:
$$\lim_{x\to \infty} x^2 \int_{0}^{x} e^{t^3 – x^3}dt$$
I've tried L'hopital's rule but the answer comes out to be $-\infty$
$$\lim_{x\to \infty} \frac{\frac{d}{dx}{\int_{0}^{x}{e^{x^3 – x^3}}}{x}}{\frac{d}{dx}\left(\frac{1}{x^2}\right)}$$
$$\lim_{x\to \infty} \frac{3x^2e^{x^3 – x^3}}{\frac{-2}{x^3}}$$
$$\lim_{x\to \infty} -\frac{3x^5}{2} = -\infty$$
Graph of the function shows limit to be not $-\infty$
Best Answer
$$\lim_{x\rightarrow \infty}\large \frac{\int^{x}_{0}e^{t^3-x^3}dt}{x^{-2}} = \lim_{x\rightarrow \infty}\frac{\int^{x}_{0}e^{t^3}dt}{x^{-2}e^{x^3}}$$
Using L'hopital's rule
$$\large \lim_{x\rightarrow \infty}\frac{e^{x^3}}{x^{-2}\cdot e^{x^3}\cdot 3x^2-e^{x^3}\cdot 2x^{-3}} = \frac{1}{3}$$