## math456 3 years ago Help.. Prove that lim as x approaches to 0. x^4cos(2/x)=0

1. math456

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2. Mikael

Look ziz iz a seempl applicashion of limit lohz:|dw:1347737182410:dw|

3. Mikael

Now , since $\lim_{x \rightarrow 0} \cos(x) = 1$

4. math456

yea

5. Mikael

And, on the other hand $\lim_{x \rightarrow 0} x^4 = 0$

6. Mikael

You have the product of those two being (surprise , surprise !) 0

7. math456

gotch yaa..!!!

8. math456

thank you!!

9. Mikael

Ahh sorry it is 2/x so here is the correct solution:

10. Mikael

Theorem we all can use states that Product of a Bounded function by another function that tends to zero - also tends to ZERO in the same limit

11. RaphaelFilgueiras

use squeeze theorem

12. Mikael

Buut$|\cos(\frac{2}{x})| \leq 1, alwayz$

13. Mikael

So we do have true the required assumptions: One of the factors (i.e. cos 2/x) is a bounded function while the other factor : x^4 tends to 0 when x-->0

14. RaphaelFilgueiras

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15. math456

yea i got it

16. Mikael

Medal ?

17. math456

thank you both!!

18. Mikael

Thanks @math456