displaystyle limx → 0 ((10/9)(sin 9x/sin 10x)) ((8/7)(sin 7x/sin 8x)) ((6/5)(sin 5x/sin 6x)) ((4/3)(sin 3x/sin 4x)) ((sin x/sin 2x)) =

Question

displaystyle limx → 0 ((10/9)(sin 9x/sin 10x)) ((8/7)(sin 7x/sin 8x)) ((6/5)(sin 5x/sin 6x)) ((4/3)(sin 3x/sin 4x)) ((sin x/sin 2x)) = (A) (63/256) (B) (1/6) (C) (6/5) (D) (1/2)

Tardigrade Admin · Accepted Answer

undersetx arrow 0 lim((10/9) ( sin 9 x/ sin 10 x))((8/7) ( sin 7 x/ sin 8 x)) ((6/5) ( sin 5 x/ sin 6 x))((4/3) ( sin 3 x/ sin 4 x))(( sin x/ sin 2 x)) = undersetx arrow 0 lim (( sin 9 x/9 x) ⋅ (1/( sin 10 x)10 x))(( sin 7 x/7 x) ⋅ (1/( sin 8 x)8 x)) =(( sin 5 x/5 x) ⋅ (1/( sin 6 x)6 x))(( sin 3 x/3 x) ⋅ (1/( sin 4 x)4 x)) (( sin x/x) (1/( sin 2 x)2 x) ⋅ (1/2)) =(1/2)[∵ lim f(x) arrow 0 ( sin f(x)/f(x))=1]

Q. $x \to 0 lim (\frac{10}{9} \frac{s in 9 x}{s in 10 x}) (\frac{8}{7} \frac{s in 7 x}{s in 8 x}) (\frac{6}{5} \frac{s in 5 x}{s in 6 x}) (\frac{4}{3} \frac{s in 3 x}{s in 4 x}) (\frac{s in x}{s in 2 x})$ =

$\frac{63}{256}$

$\frac{1}{6}$

$\frac{6}{5}$

$\frac{1}{2}$

$\frac{256}{63}$

Q. x→0lim​(910​sin10xsin9x​)(78​sin8xsin7x​)(56​sin6xsin5x​)(34​sin4xsin3x​)(sin2xsinx​) =

25663​

61​

56​

21​