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What is the limit of #x(cos(1/x))# as x approaches infinity?

Answer 1

Cameron Alexander

#lim_(xrarroo)x(cos(1/x)) = oo#

As #x# increases without bound, #1/x rarr 0#.

The cosine function is continuous at #0#, so

#lim_(xrarroo)cos(1/x) = cos0 = 1#.

We can conclude that, as #x# increases without bound, #xcos(1/x)# also increases without bound.

That is, #lim_(xrarroo)x(cos(1/x)) = oo# .

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Answer 2

Adam Adkins

The limit of x(cos(1/x)) as x approaches infinity is 0.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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