Final answer:
To find the limit of sin ax / sin bx as x approaches 0, we apply L'Hôpital's Rule by differentiating the numerator and the denominator to get a/b.
Step-by-step explanation:
We are asked to evaluate the limit lim(x→0) sin ax / sin bx, where a, b ≠ 0. To tackle this problem, we can apply L'Hôpital's Rule which allows us to evaluate limits of indeterminate forms by differentiating the numerator and denominator separately.
As x approaches 0, both sin ax and sin bx approaches 0, making it an indeterminate form 0/0. Therefore, we differentiate the numerator and denominator with respect to x:
The derivative of sin ax with respect to x is a cos ax, and the derivative of sin bx with respect to x is b cos bx. Substituting these into the original limit, we get lim(x→0) a cos ax / b cos bx.
As x approaches 0, cos ax and cos bx both approach 1. Thus, the limit simplifies to a/b. Hence, the final answer to the limit is a/b.