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 ∞/∞).

Comment [5]

Limit of Exponential

Posted Fri May 2, 2008 in

Young Son brought me a Calculus problem the other night. The problem was to compute the limit lim (h->0) (ex+h-ex)/h. This is the definition of the derivative of the exponential with respect to x and the answer is clearly ex.

However, proving this is not so trivial! I cheated a looked it up and realized <slaps head> that an ex factors from the numerator. The ex also is independent of h, so it comes outside the limit. That leaves lim (h->0) (eh-1)/h.

Hmm…

I haven’t found a rigorous proof of the result, but by successive approximation the limit converges to unity. What an interesting problem. I’d really like to see an elegant proof. Well, even better would be if I constructed such an elegant proof independently. :)

In working through this little bit of mathematics, I came across The Math Page and in particular this entry. This site is worth a visit.

Comment [9]

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