Perturbation theory and asymptotics

[Juggler123]

I need to find the roots of the transcendental function,

f(x;a)=x^2-3ax-1-a+exp(-x/a)=0;

I've done many problems like this before and am fairly sure this is just a regular perturbation problem. The difficulty I'm having is with the exponential term.

Could anyone give me an idea of how to tackle this problem?

Thanks.

 

[HallsofIvy]

Approximate the exponential with a Taylor's polynomial:
$$exp(-x/a)= 1- x/a+ \frac{x^2}{2a^2}+ \cdot\cdot\cdot+ (-1)^n\frac{x^n}{n!a^n}$$

 

[Juggler123]

Sorry I didn't make it clear in my first post, I'm finding the roots of the equation as a tends to 0. I thought about the Taylor expansion but then the terms are being divided by 0 (as a tends to zero). Is there anyway around this?

Thanks.