Limit of Exponential, Reprise

Posted Tue May 27, 2008 in

It’s Tuesday afternoon and I have a short break before going on to the next thing. There’s always a next thing.

Friday we head out for Texas. I think I’m ready for some road time. I like the driving experience and the opportunity to think is welcome. We should be in Lubbock before Sunday, I hope.

I was thinking about Jim’s comments about my Limit of the Exponential hack. I am still wondering how to prove it analytically. I can do it numerically, but that’s not the same. L’Hopital’s1 rule applies, but that seems like cheating.

If you examine the function f(h)=(eh – 1)/h as h approaches zero from either the positive or negative size of zero, then the limit approaches unity. However, the function is undefined at h=0. Therefore, the function is discontinuous at h=0, but the limit exists and is unity. This is really pretty cool.

I’ll see if I can construct a graph of the function and insert it here later. Now back to work.

1 L’Hopital’s rule applies to derivatives of the indefinite forms (0/0 and ∞/∞).