solve the limit of the function f(x)=sin5x/sinx if x --> pi
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We have to find the value of lim x--> pi[ sin 5x / sin x]
We see that substituting x with pi gives us the form 0/0 which is indeterminate. We can use therefore use l'Hopital's rule and use the derivative of the numerator and the denominator
lim x--> pi [sin 5x / sin x]
=> lim x--> pi [ 5* cos 5x / cos x]
substtuting x = pi , now gives
(5*-1)/ (-1)
=> 5
The required value of lim x--> pi[ sin 5x / sin x] is 5.
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We'll create the remarcable limits:
lim sin x/x = 1, if x->0
We'll re-write the function:
lim [(sin 5x)/5x]*[(5x)/sin x] = lim [(sin 5x)/5x]*lim [(5x)/sin x]
lim [(sin 5x)/5x]*lim [(5x)/sin x] = 1*lim [(5x)/sin x]
1*5lim [(x)/sin x] = 1*5 = 5
The limit of the given function is : lim sin5x/sinx = 5, if x -> pi.
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