TITLE: Adding constant to the variable in a Limsup
QUESTION [5 upvotes]: Let $f(t)$ be a real valued function and $C$ a constant. Is it true that
$$
\limsup_{t\to\infty}\frac{f(t)}{t}=\limsup_{t\to\infty}\frac{f(t+C)}{t} ?
$$
I have tried to prove it using a change of variable $s=t+C$, but the denominator seems problematic.

REPLY [12 votes]: Hint:
$$
\frac{f(t+C)}{t}=\frac{f(t+C)}{t+C}\,\frac{t+C}{t}.
$$