evaluate lim x=0 (Evaluate sin 3x cot 5x/xcot 4x)

Answer 1

Using L'Hopital's rule:

lim x→0 [sin 3x cot 5x/xcot 4x]
= lim x→0 [(3cos 3x cosec^2 5x - 5cosec^2 5x sin 3x)/(cot 4x - xcosec^2 4x)]
= [3cos 0 cosec^2 0 - 5cosec^2 0 sin 0]/cot 0
= 3/1 = 3

Therefore, lim x→0 (sin 3x cot 5x/xcot 4x) = 3.