One of my colleagues placed the following problem on an exam for his Calculus II course…
and was impressed by the variety of correct responses that he received. I thought it would be fun to discuss some of the different ways that this limit can be computed.
Method #2. Using L’Hopital’s Rule. The limit has the indeterminant form , and so I can differentiate the top and the bottom with respect to :
Oops… it looks like I just got the reciprocal of the original limit! Indeed, if I use L’Hopital’s Rule again, I’ll just return back to the original limit.
So that doesn’t look very helpful… except it is. If I define the value of this limit to be equal to , then I’ve just shown that (assuming that the limit exists in the first place, of course). That means that or . Well, clearly the limit of this nonnegative function can’t be negative, and so we conclude that the limit is equal to .