Let `f(x)= cosec 2x + cosec 2^2 x+ cosec 2^3 x+........+ cosec 2^n ,x in (0,pi/2) and g(x)=f(x)+cot 2^n x` . If `H(x)={ (cosx)^(g(x))+(sec)^(cisecx) if x lt 0 and p if x=0 and (e^x+e^(-x)-2cosx)/(x sin x) If x lt o` .Find the value of p, if possible to make the functieIf `H(x)` continuous at `x = 0`.

Options

A. `(1)/(2)`
B. 1
C. 2
D. 0

Answer

A

Explanation

`f(x)="cosec"2x+"cosec"2^(2)x+ . . . . . . .+"cosec"2^(n)x`
now cosec `2x=(1)/(sin2x)=(sin(2x-x))/(sinxsin2x)=cotx-cot2x`
`|||ly" cosec"2^(2)x=cot2x-cot2^(2)x`
`"cosec"2^(3)x=cot2^(2)x-cot2^(2)x-cot2^(3)x`
`vdots`
` ul("cosec"2^(n)x=cot2^(n-1)x-cot2^(n)x)`
`:.f(x)=cotx-cot2^(n)x`
`:.g(x)=f(x)+cot2^(n)x=cotx`
now `H(0+h)=underset(hto0)Lim((cos""h)^(cot""h)+(sec""h)^("cosec h"))=e^(overset(Lim coth (cosh-1))overset(hto0" ")" "+=e^(overset(Lim cosech(sech-1))overset(hto0" ")`
=1+1=2 . . . . (1)
`H(0-h)=underset(hto0)Lim(e^(-h)+e^(h)-2cosh)/(hsin""h)=underset(hto0)Lim{(e^(h)+e^(-h)-2)/(h^(2))+(2(1-cosh))/(h^(2))}=2`
From (1) and (2) h(x) will be cont. if p=2