Doesn't look particularly difficult or interesting to me. Just apply L'Hopital's rule twice.
Well, having forgotten all about L'Hôpital's rule, I thought it was interesting. And I think a student taking calculus might have found it interesting. Oh well, ebaines is right:
I see there's a tough crowd here. My apologies for bringing a butter knife to a gunfight.
Even though the problem is easy to solve with the use of L'Hôpital's rule, from an intuitive perspective I still find the result interesting.
While testing an approach to another problem, I started with the basic function:
I wanted to modify it so that it would satisfy , and came up with:
Then I wanted to modify it further so that it would intersect , and came up with:
Finally, I wanted to add an argument to vary the curvature of the graph while maintaining the preceding requirements, so I changed it to:
While playing around with different values of in a graphing app, I discovered (graphically, rather than analytically) that the function collimated with when . That was completely unexpected, and, I thought, intriguing… regardless of how elementary the analytic proof is.