Evaluate the limits of x^4*cos(2/x). x-->0

We'll  notice that we can't use the product law here (the
limit of a product is the product of limits).

According to
the rule, limit of cos(2/x) does not exist, if x tends to
0.

By definition:

 -1
=< cos (2/x) =< 1

If we'll multiply the
inequality above, by x^4, because x^4 is a positive amount, for any value of x, the
inequality still holds.

-x^4 =< (x^4)*cos (2/x)
=< x^4

We'll calculate the limits of the
ends:

If we'll calculate lim x^4 = lim -x^4 =
0.

Now, we'll apply the Squeeze Theorem and we'll get
:

lim -x^4 =< lim (x^4)*cos (2/x) =< lim
x^4

0=< lim (x^4)*cos (2/x)
=<0

So, the  limit of the
function (x^4)*cos (2/x) is 0, when x ->
0.