1.

If Lim_(xto oo) "sin"((pi(1-cos^(m)x))/(x^(n))) exists and non-zero where m,n in N then

Answer»

m=1 , n=1
m=1 , n=2
m=2 , n=2
m=3 , n=2

Solution :When m=1, n=1
`UNDERSET(NTO oo)Lim"SIN"((pi(1-cosx))/(x))=sin0=0`
When m=1 , n=2
`underset(nto oo)Lim"sin"((pi(1-cosx))/(x^(2)))=sin(pi//2)=1`
When m=n=2
`underset(nto oo)Lim"sin"((pi(1-cos^(2)x))/(x^(2)))=underset(nto oo)Lim"sin"((pi(1-cosx)(1+cosx))/(x^(2)))=sin(pi)=0`
When m=3 , n=2
`underset(nto oo)Lim"sin"((pi(1-cos^(3)x))/(x^(2)))=underset(nto oo)Lim"sin"((pi(1-cosx)(1+cos^(2)x+cosx))/(x^(2)))=sin((3PI)/(2))=-1`


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