Question

If x is positive integer and n^x > x, then is n^infinity > infinity?

Answer 1

In general, taking a limit can only give you a weak inequality ≥, not a strict inequality >. For example, it is true that 1/x^(2)>0 everywhere, but in the limit you get lim 1/x = 1/infinity = 0, not >0.

Likewise, even if n^(x)>x everywhere, in the limit you only get n^(∞)≥∞, not strictly greater. Out of the two possibilities "greater than or equal to", this one is equal to.