Explanation

limₓ→0 (sin(ax) / sin(bx))

Divide the numerator and the denominator by x:

= limₓ→0 [(sin(ax)/x) ÷ (sin(bx)/x)]  

= (a/b) × limₓ→0 [sin(ax)/(ax)] ÷ [sin(bx)/(bx)]

As x → 0, sin(kx)/(kx) → 1, so:

Answer: a / b