1.

If lim_(xrarr0) (f(x))/(x^(2))=a and lim_(xrarr0) (f(1-cosx))/(g(x)sin^(2)x)=b (where b ne 0), then lim_(xrarr0) (g(1-cos2x))/(x^(4)) is

Answer»

`(4a)/(b)`
`(a)/(4b)`
`(a)/(b)`
none of these

Solution :`underset(xrarr0)(LIM)(F(1-cosx))/(g(X)sin^(2)x)`
`""=underset(xrarr0)(lim)(f(2SIN^(2).(x)/(2)))/(g(x)(2sin^(2).(x)/(2)))xx((2sin^(2).(x)/(2))^(2))/(4(sin^(2)(x)/(2))(cos^(2).(x)/(2)))`
`""=underset(xrarr0)(lim)(a)/(g(x))xxtan^(2).(x)/(2)`
`""a underset(xrarr0)(lim)(((x)/(2))^(2))/(g(x))xx(tan^(2).(x)/(2))/(((x)/(2))^(2))`
`""=a underset(xrarr0)(lim)(x^(2))/(4g(x))`
`therefore""underset(xrarr0)(lim)(x^(2))/(g(x))=(4b)/(a)`
Now, `underset(xrarr0)(lim)(g(2sin^(2))x)/(x^(4))`
`""underset(xrarr0)(lim)(g(sin^(2)x))/((2sin^(2)x)^(2))xx((2sin^(2)x)^(2))/(x^(4))`
`""=(a)/(4b)xx4=(a)/(b)`


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