Use Theorem 3.10 to evaluate the following limits.
lim x🠂0 sin ax / sin bx, where a and b are constants with b ≠ 0.

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Theorem 3.10 is the standard limit result: lim x→0 (sin x)/x = 1. This theorem is useful for evaluating limits involving sine functions as x approaches 0.

To evaluate lim x→0 (sin(ax)/sin(bx)), we can rewrite it as (sin(ax)/ax) * (bx/sin(bx)) * (a/b).

Apply Theorem 3.10 to each sine term: lim x→0 (sin(ax)/ax) = 1 and lim x→0 (bx/sin(bx)) = 1.

Combine the results from the application of Theorem 3.10: lim x→0 (sin(ax)/sin(bx)) = (1) * (1) * (a/b).

Thus, the limit evaluates to a/b, which is the final expression for the given limit problem.

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Theorem 3.10 typically refers to a limit theorem that helps evaluate the limit of a quotient of functions. In this case, it likely states that if the limits of the numerator and denominator both approach zero, the limit of their quotient can be evaluated using L'Hôpital's Rule or by simplifying the expression. Understanding this theorem is crucial for solving limits that result in indeterminate forms.

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