Given $f(x)$ is a continuous function defined in $(0,\: \infty)$ such that $$\lim_{x \to \infty}f(x)=1$$ Then Find $$L=\lim_{x \to \infty}{\mathrm{e}^{-x}\int_{0}^{x}{f\left(y\right)\mathrm{e}^{y}\,\mathrm{d}y}}$$
My try:
we have $$L=\lim_{x \to \infty}{\frac{\int_{0}^{x}{f\left(y\right)\mathrm{e}^{y}\,\mathrm{d}y}}{e^x}}$$
If the numerator is finite then $L=0$ else by L'Hopital's Rule we have $\infty/\infty$ form we get
$$L=\lim_{x \to \infty}\frac{f\left(x\right)\mathrm{e}^{x}}{\mathrm{e}^x}=\lim_{x \to \infty}{f\left(x\right)}=1$$
But how to tell whether $\lim\limits_{x \to \infty}\int_{0}^{x}{f\left(y\right)\mathrm{e}^{y}\,\mathrm{d}y}$ is Finite or Infinite?
Best Answer
Hint: Since $\lim_{x\to\infty}f(x)=1$, we have $f(x)>1/2$ for all $x>N$, where $N$ is sufficiently large. Ignoring the integral over $[0,N]$ (which is a fixed finite value), we consider the remaining part: $$\int_N^\infty f(x)e^xdx\geq\frac{1}{2}\int_N^\infty e^xdx$$