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.

Verified step by step guidance

1. 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.
2. To evaluate lim x→0 (sin(ax)/sin(bx)), we can rewrite it as (sin(ax)/ax) * (bx/sin(bx)) * (a/b).
3. Apply Theorem 3.10 to each sine term: lim x→0 (sin(ax)/ax) = 1 and lim x→0 (bx/sin(bx)) = 1.
4. Combine the results from the application of Theorem 3.10: lim x→0 (sin(ax)/sin(bx)) = (1) * (1) * (a/b).
5. Thus, the limit evaluates to a/b, which is the final expression for the given limit problem.

Key Concepts

Theorem 3.10 (Limit Theorem)

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.

Sine Function Behavior Near Zero

The sine function has a well-known behavior as it approaches zero, specifically that lim x→0 (sin x)/x = 1. This property is essential for evaluating limits involving sine functions, as it allows us to rewrite expressions in a more manageable form. Recognizing how sine behaves near zero is key to simplifying the limit in the given problem.

Constant Multiplication in Limits

When evaluating limits, constants can be factored out of the limit expression. This means that if a limit involves constants multiplied by functions, those constants can be treated separately. In the context of the given limit, understanding how to handle the constants 'a' and 'b' will help simplify the expression and arrive at the correct limit value.