If a multivariable continuous real function has $\lim_{\Vert x\Vert\to\infty} f(x) = L$, it is then uniformly continuous?

Question

An exercise sheet of my Multivariable Calculus subject asks you to prove that a continuous function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$ is uniformly continuous if $\lim_{\Vert x\Vert\to\infty} f(x) = L$. I think I have nevertheless found a counterexample: taking $n=1$, the function $f(x)=\sin(1/x)$ for $x\neq 0$, $f(0)=0$ has limit 0 as x approaches both negative and positive infinity, but it is not uniformly continuous. Is my reasoning correct? Otherwise, can you give me some hints on how to deal with this proof? Thanks in advance.