Limit of Exponential
Posted Fri May 2, 2008 in
Mathematics
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.
I don’t know about me sometimes. I’ll take a “wild hair” and write an article like the one above, then a friend will counter-post this. <chuckles> He’s absolutely correct about me, of course, but I didn’t think I was that transparent!
— ruminator 2 May 2008, 08:48 #I disagree completely. I find it very stimulating.
— Dazed n Confused 2 May 2008, 14:40 #Obviously it goes to the indeterminate form 0/0. What does L’Hopital’s rule say?- I think it’s 1.
I asked Young Son if he had received instruction on L’Hopital’s rule, but it doesn’t come for several chapters in his text. Therefore, it should be approachable from first principles. Yet, I don’t see the solution to the problem. Empirically I can get there and I can also approach the problem geometrically (there is no asymptote at h=0, therefore the limit exists and is the same from both sides) (the derivative of the limit function is continuous).
Fascinating problem.
— ruminator 2 May 2008, 15:10 #I reiterate: The answer is clearly have a muffin and a cup of tea and relax. :)
— Jim 2 May 2008, 19:31 #I like muffins… I like tea… However, a couple of weeks ago we had an interesting thread via company email and it seems I might as well eat Krispy Kreme doughnuts as eat a Costco muffin. There is less total fat in three Krispy Kremes than in a single Costco cocoa muffin. Besides, doughnuts and coffee are so much more satisfying than muffins…
Hmm… The first line sounds like something from an old Michael Oldfield album…
— ruminator 2 May 2008, 20:12 #Yes. But I fear the sugar rush from the donuts might get you all excited about mathematics again. A muffin is going to slow you down. How many times have I felt like I needed a nap after eating a big ol’ muffin?
— Jim 3 May 2008, 14:46 #Honestly, I never have experienced much sensitivity to sugar. I don’t add much sugar to my foods or drinks (sometimes a bit of Splenda to my coffee for a treat), or a little bit on my cereal (rarely). I use honey more often as a sweetener, if I desire something sweet.
Candy bars and other prime sugar sources don’t give me a rush. At least, they don’t that I know of.
I can understand, though, how you might feel the need for a nap after a Costco muffin — the amount of processing your body has to do on the fats would make anyone sleepy. I’m surprised the bears don’t raid the Costco in search of muffins!
— ruminator 3 May 2008, 14:50 #I recently read an article in Science Magazine or Scientific American that proved that Costco muffins have a weak, but influential gravitational field. I can’t seem to find the source online right now, but if I come across it I’ll post up a link.
— Jim 3 May 2008, 15:19 #Based on Newton’s work, everything has a gravitational field. However, such gravitational field might not have sufficient strength to be measurable.
Your gravitational field might not be measurable, but I’d wager that mine is! ;)
That being said, I think I saw an article in the Journal of Irreproducible Results about a set of instruments designed to measure the gravitational field of a Costco muffin and an experiment (and the resulting data) to prove it. (See JIR.)
— ruminator 5 May 2008, 04:57 #