prove lim x--> 0 x^4 cos 2/x = 0

Question

prove lim x--> 0   x^4 cos 2/x = 0

Answer 1

Using the Squeeze Theorem, lim cos (2/x) (as x goes toward zero) is no more than 1 and no less than -1, simply because it's a cosine of something.

 

Then the limit of the product being the product of individual limits, your overall problem becomes 0 times some number in the interval [-1,1], which is zero in any case.  Q.E.D.