About convergence of logs

Question

I have a function

$f(c,x)=\log(c^x/x)$

I want to know the limit of this function when $c$ is constant, but $x \to \infty$.

Simulations suggest it tends to infinity, but I'd like to have a formal argument.

Answer 1

Note that

• $0<c\le1\implies c^x/x\to0\implies f(c,x)=\log(c^x/x)\to -\infty$

• $c>1\implies c^x/x\to \infty\implies f(c,x)=\log(c^x/x)\to +\infty$