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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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.