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:

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

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