Tough calculus problem

Here we go, I think I got the point where E^t overtakes T^N for all N so people can zoom out their graphs far enough to see what's going on. E^T = T^N where N = Ln(T/Ln(T)). Obviously we'd rather have this as T = something of N still rejiggering with it, been a while since I took calculus.

So, it's clear that e^t approaches infinity as t goes to infinity, and it's clear that -t^n approaches negative infinity as t goes to infinity. Since it's a case of infinity - infinity, it's not entirely clear which function dominates the other. If e^t dominates, the function will go to infinity, if -t^n dominates, the function will go to negative infinity. You can test to see which function dominates by using L'Hôpital's rule, simply put e^t / t^n and evaluate what the limit of that is. If the limit approaches infinity, then e^t dominates. If the limit approaches 0, then -t^n dominates. The rest is up to you.

E^t outgrows t^n for any real n. Infinity.

For 1 [u]<[/u] n [u]<[/u] 6, the limit is infinity For n > 6, the limit is negative infinity

Calculus one or two?