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

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neela | High School Teacher | (Level 3) Valedictorian

Posted on

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.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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.

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