Question:

r+1 3 r+1 tan(x/2 ) + tan (x/2 ) f (x) = lim ( ) (1) x\to\infty 1 - tan (x/2 2 r+1 ) x tan( ) r+1 2 = lim x\to\infty x cos( r ) 2 r+1 sin(x/2 ) = lim x\to\infty x x cos( ) cos( r ) r+1 2 2 x x sin( r - ) 2 r+1 2 = lim x\to\infty x x cos( ) cos( r ) r+1 2 2 x x = limx\to\infty tan( r ) - tan( ) 2 r+1 2 From condition given question n x x \therefore lim \sum [tan( ) - tan( )] = tan x r r+1 n\to\infty 2 2 r=0 x tan x e - e \therefore lim ( ) x\to0 x - tan x x-tan x e - 1 tan x \Rightarrow lim e ( ) x\to0 x - tan x x-tan x e - 1 tan x \Rightarrow lim e lim ( ) x\to0 x\to0 x - tan x x-1 e \Rightarrow 1.1 (∵ lim = 1) x\to0 x = 1