This question may be as much computer science as math, as I don't believe the reason is mathematical; however an odd thing a classmate of mine pointed out: if you look at a graph of the function in question on any kind of graphing calculator, and get really really REALLY close to 0, the graph starts oscillating(ex:http://m.wolframalpha.com/input/?i=graph+y+%3D+%28e%5Ex-1%29%2Fx+from+-1x10%5E-14+to+1x10%5E-14&x=-1244&y=-71)
I was wondering if anyone could offer an explanation as to why this phenomenon occurs.
If you take your function on Wolfram Alpha, try replacing $-14$ by $-15$. You will see that Wolfram Alpha evaluates at a few discrete points, then connects the dots in some way. Hence, your viewing window is right up against the limits of computation, and rounding error (as @hardmath describes in more detail in the comments) becomes amplified.