The following is from a book of PDEs and I have cannot seem to figure out a particular step in it with regard to the derivation of the fundamental solution of the heat equation. I have highlighted it in red. I would much appreciate if someone could help me out with it or if possible suggest me a text which has a more detailed approach than what I have attached. Thanks!
$$\begin{align} e^{-r}\int^r e^\sigma\,\sigma^{-1/2}\,d\sigma&= e^{-r}\Bigl(e^\sigma\,\sigma^{-1/2}\Bigr|^r+\frac12\int^re^\sigma\,\sigma^{-3/2}\,d\sigma\Bigr)\\ &=r^{-1/2}+\frac{e^{-r}}{2}\int^re^\sigma\,\sigma^{-3/2}\,d\sigma\\ &=r^{-1/2}+O(r^{-3/2}), \end{align}$$ since $$ \lim_{r\to\infty}\frac{\int^re^\sigma\,\sigma^{-3/2}\,d\sigma}{r^{-3/2}e^r}=\lim_{r\to\infty}\frac{e^r\,r^{-3/2}}{(-(3/2)r^{-5/2}+r^{-3/2})e^r}=\lim_{r\to\infty}\frac{1}{-(3/2)r^{-1}+1}=1. $$