Final answer:
The limit of sin(ax) / sin(bx) as x approaches 0, given a and b are nonzero, is evaluated using L'Hôpital's rule. As x becomes 0, it simplifies to the constant ratio a/b.
Step-by-step explanation:
The question asks us to evaluate the limit of the function sin(ax) / sin(bx) as x approaches 0, where a and b are nonzero constants. To find this limit, we can use L'Hôpital's rule since direct substitution of x = 0 would result in the indeterminate form 0/0.
By L'Hôpital's rule, we can differentiate the numerator and the denominator separately with respect to x and then take the limit as x approaches 0. The derivatives are a cos(ax) for the numerator and b cos(bx) for the denominator. After differentiating, we get:
lim x→0 (sin(ax) / sin(bx)) = lim x→0 ((a cos(ax)) / (b cos(bx))) = (a/b).
We can see that the limit does not depend on x since the cos(ax) and cos(bx) terms will equal 1 when x is substituted with 0, leaving us with the constant ratio a/b.