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math456

  • 3 years ago

Help.. Prove that lim as x approaches to 0. x^4cos(2/x)=0

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  1. math456
    • 3 years ago
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    |dw:1347737098765:dw|

  2. Mikael
    • 3 years ago
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    Look ziz iz a seempl applicashion of limit lohz:|dw:1347737182410:dw|

  3. Mikael
    • 3 years ago
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    Now , since \[\lim_{x \rightarrow 0} \cos(x) = 1\]

  4. math456
    • 3 years ago
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    yea

  5. Mikael
    • 3 years ago
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    And, on the other hand \[\lim_{x \rightarrow 0} x^4 = 0\]

  6. Mikael
    • 3 years ago
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    You have the product of those two being (surprise , surprise !) 0

  7. math456
    • 3 years ago
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    gotch yaa..!!!

  8. math456
    • 3 years ago
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    thank you!!

  9. Mikael
    • 3 years ago
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    Ahh sorry it is 2/x so here is the correct solution:

  10. Mikael
    • 3 years ago
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    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
    • 3 years ago
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    use squeeze theorem

  12. Mikael
    • 3 years ago
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    Buut\[|\cos(\frac{2}{x})| \leq 1, alwayz\]

  13. Mikael
    • 3 years ago
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    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
    • 3 years ago
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    |dw:1347737519682:dw|

  15. math456
    • 3 years ago
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    yea i got it

  16. Mikael
    • 3 years ago
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    Medal ?

  17. math456
    • 3 years ago
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    thank you both!!

  18. Mikael
    • 3 years ago
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    Thanks @math456

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