Uniform convergence of $\frac{f(xt)}{f(t)}$ as $t\to\infty$

Question

Let $f(x)$ decrease monotonically to $0$ as $x\to\infty$.

Suppose $\lim\limits_{t\to\infty}\frac{f(xt)}{f(t)} = x^p$ for all $x>0$ and some $p<0$.

Can this convergence be nonuniform in $x$?

Answer 1

On $(1,\infty)$ the function $f(x)=\frac 1 {\ln\, x}$ is decreasing and $\frac {f(xt)} {f(t)} \to 1$ as $t \to \infty$. But this convergence is not uniform as seen by putting $x=t$. [Here $p=0$].