How do you use the Squeeze Theorem to find #lim xcos(1/x)# as x approaches zero?

1 Answer
Bill K.
Nov 13, 2015

Use the fact that the cosine function is always between #-1# and #1#, implying that the given function is always between #-|x|# and #|x|#, which both go to zero as #x# goes to zero.

Explanation:

Let #f(x)=x cos(1/x)#, #g(x)=-|x|#, and #h(x)=|x|#. Since #-1 leq cos(1/x) leq 1# for all #x !=0#, it follows that #g(x) leq f(x) leq h(x)# for all #x !=0#.

But #lim_{x->0}g(x)=lim_{x->0}h(x)=0#. Therefore, the Squeeze Theorem can be use to conclude that #lim_{x->0}f(x)=lim_{x->0}x cos(1/x)=0#.

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