math456
Help.. Prove that lim as x approaches to 0. x^4cos(2/x)=0
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math456
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|dw:1347737098765:dw|
Mikael
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Look ziz iz a seempl applicashion of limit lohz:|dw:1347737182410:dw|
Mikael
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Now , since \[\lim_{x \rightarrow 0} \cos(x) = 1\]
math456
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yea
Mikael
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And, on the other hand \[\lim_{x \rightarrow 0} x^4 = 0\]
Mikael
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You have the product of those two being (surprise , surprise !)
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math456
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gotch yaa..!!!
math456
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thank you!!
Mikael
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Ahh sorry it is 2/x so here is the correct solution:
Mikael
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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
RaphaelFilgueiras
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use squeeze theorem
Mikael
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Buut\[|\cos(\frac{2}{x})| \leq 1, alwayz\]
Mikael
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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
RaphaelFilgueiras
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|dw:1347737519682:dw|
math456
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yea i got it
Mikael
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Medal ?
math456
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thank you both!!
Mikael
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Thanks @math456