Consider the function `f(x)=lim_(nto infty) ((1+cosx)^(n)+5lnx)/(2+(1+cosx)^(n))`, then

Options

A. `f(x)` is discontinous at positive even multiples of `pi`
B. `f(x)` is continuous at positive odd multiples of `pi`
C. `f(x)` is discontinuous at positive odd multiples of `(pi)/(2)`
D. `f(x)` is continuous at positive even multiples of `(pi)/(2)`

Answer

A, C, D

Explanation

`f(x)={{:(1,,, x in(20npi,2npi+(pi)/2),n in I^(+)cup{0}),((1+5lnx)/3,,,x=(4n+1)(pi)/2,n in I^(+)cup{0}),((5lnx)/2,,,x in(2npi+(pi)/2,2npi+(3pi)/2),n in I^(+_)cup{0}),((1+5lnx)/3,,,x=2npi+(3pi)/2, n in I^(+)cup{0}):}`
1: `x in (2npi+(3pi)/(2),2npi+2pi)n in I^(+)cup{0}`
`f(x)` is discontinuous at all positive odd multiples of `(pi)/(2)`