Question

how to do evaluate .. lim x-0 (1-cos4x)/(1-cos5x)

Answer 1

lim      (1-cos4x/1-cos5x) 
x-->0 

lim 
x-->0 [2 sin square (2x)/ 2 sin square (5x/2)] 

lim 
x-->0 [sin (2x) . Sin(2x)/ sin (5x/2) . Sin (5x/2] 

lim 
x-->0  [(sin2x/2x) (sin 2x/2x) / (sin {5x/2}/5x/2) (sin{5x/2}/5x/2)]  x 4x square / 25x square /4 

= lim x-->0(sin2x/2x) lim x--> 0 (sin 2x/2x) / lim x --->0(sin {5x/2}/5x/2) lim x-->0 (sin{5x/2}/5x/2) x 16/25 

1x1/1x1 X 16/25 [ as, lim x-->0 sinx/x = 1 ] 

= 16/25