Adding constant to the variable in a Limsup

Question

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.

Why does the denominator seem problematic? $t-C\to \infty$ as $t\to\infty$, — Jimmy R., Nov 15 '16 at 10:07
Because when I substitute I no longer have $t$ on the denominator but $t-C$. — Sak, Nov 15 '16 at 10:08

score 12 · Accepted Answer · answered Nov 15 '16 at 10:07

12

Hint: $$ \frac{f(t+C)}{t}=\frac{f(t+C)}{t+C}\,\frac{t+C}{t}. $$

answered Nov 15 '16 at 10:07

Julián Aguirre

THank you! I have been able to come up with a proof using this. This is a very clever trick. – Sak Nov 15 '16 at 10:24

1 Answers1