$lim(x->0)$ sin(x)*sin( $1/x$ ) solve the limit?

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Answer 1 · 2018-07-19T14:29:37

$0$

Notice, that $-1\le \sin\theta\le 1\ \forall \ \ \theta\in \mathbb R$

$\therefore -1\le \sin(1/x)\le 1\ \forall \ \ x\in \mathbb R$

Given that

$\lim_{x\to 0}\sin x\cdot \sin(1/x)$

$=\lim_{x\to 0}\sin x\cdot \lim_{x\to 0}\sin(1/x)$

$=0\cdot (k)\ \quad (\text{where}, \ -1\le k\le 1)$

$=0$

lim(x->0)lim(x->0) sin(x)*sin(1/x1/x) solve the limit?