How do you find #lim_(x to 0) xcos(1/x)#?

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Answer 1 · 2017-08-28T22:04:45

Given: #lim_(x to 0) xcos(1/x)#

We know that the cosine function is bounded by -1 and 1:

#-1 < lim_(x to 0) cos(1/x) < 1#

And we know that

#lim_(x to 0) x = 0#

Therefore, the limit of a bounded function mulitplied by a function that goes to 0 must be 0:

#lim_(x to 0) xcos(1/x) = 0#