#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#