If $\lim_{x\to 0}f(x)=\infty$ and $f(x)=g(x)h(x)$, then can it be expressed as $\lim_{x\to 0}f(x)=\lim_{x\to 0}g(x)\lim_{x\to0}h(x)$?
2026-04-24 00:56:46.1776992206
if $\boldsymbol{f(x)}$ is not convergent, can $\boldsymbol{f(x)}$ be separated in limit?
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No. Suppose, for instance, that $g(x)=x$ and that $h(x)=2+\sin(x)$. Then $\lim_{x\to+\infty}f(x)=+\infty$, but the limit $\lim_{x\to+\infty}h(x)$ doesn't exist.