Does writing $f(x)\sim \ell$ have a sense?

Question

If $\lim_{x\to a}f(x)=\ell$, is it correct to say that $f(x)\sim_a \ell$ ? I would say yes since $\lim_{x\to a}\frac{f(x)}{f(a)}=1$, but on a test I wrote $e^{-t}\sim_0 1$ and the corrector said that it's wrong to say that. So, what's wrong ?

Answer 1

Yes I'd say it's true. From the property of the limit we get $|f(x)-L|<\varepsilon$, so there are two bounds, say $L-\frac{1}{n}, L+\frac{1}{n}$ such that $f(x)$ is within these bounds. Hence the ratio is 1 by squeeze lemma.