To find the limit of x^4cos(2/x) as x --> 0.
We know that |cos (2/x)| < 1.
Therefore -1 < = cos(2/x) < = 1.
WE multiply by x^4 :
-x^4 < = x^4 cos(2/x) < = x^4.............................(1)
Take x^4 cosx(2/x) < = x^4
Lt x--> 0 x^4 cos (2/x) < Lt x-->0 x^4 = 0...........(2)
Take the left side of the the inequality (1):
-x^4 < x^4 cos (2/x)
We take the limit as x--> 0.
Lt--> 0 -x^4 < = lt x--> 0 x^3 cos (2/x).
But Lt x-->0 -x^4 = 0............(3)
From (2) and (3) we conclude that Lt x--> 0 x^4 cos(2/x) , being between upper and lower bounds . But both upper and lower bounds approach zero as x--> 0. So Lt x--> 0 x^4cos(2/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.
-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.