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.
-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.