# Asymptotic expansion

1. Oct 26, 2012

### the_kid

1. The problem statement, all variables and given/known data

Find the first two terms in an asymptotic expansion of the following as x goes to zero from the right (i.e. takes on smaller and smaller positive values).

$\int^{1}_{0}$e$^{-x/t}$dt

2. Relevant equations

3. The attempt at a solution
I'm not exactly sure how to proceed with this. I'm assuming I should expand the integrand as a Taylor series:

e$^{-x/t}$=$\sum^{\infty}_{k=0}$$\frac{(-x/t)^{k}}{k!}$

I'm kinda stuck here, though.

2. Oct 26, 2012

### Zondrina

Notice you start having problems as t goes to zero... you should re-write your integral with a limit and then I think it would be best to actually expand the terms of your sum this time.

3. Oct 26, 2012

### the_kid

lim$_{a\rightarrow0^{+}}$$\int^{a}_{0}$(1-$\frac{x}{t}$+$\frac{x^{2}}{2t^{2}}$-$\frac{x^{3}}{6t^{3}}$+$\cdots$)dt

Like this? Hmm...