Question

Use Theorem 3. 10 to evaluate the following limits. $$\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}, \text { where } a \text { and } b \text { are constants with } b \neq 0$$

Short answer

Question: Find the limit of the given function as x approaches 0: \(\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}\), where a and b are constants and b is not equal to 0. Answer: The limit of the given function as x approaches 0 is 1.

Step-by-step solution

Write down the given function

The given limit function is: $$\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}$$

Apply the limit properties

We know from the theorem that \(\lim_{x\to 0} \frac{\sin(x)}{x} = 1\). Therefore, we need to manipulate the given function to fit this form. Rewrite the limit using limit properties: $$\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x} = \frac{\lim_{x \rightarrow 0} \sin a x}{\lim_{x \rightarrow 0} \sin b x}$$

Introduce the required expressions to fit the theorem

To apply the theorem, we need a term of the form \(\frac{\sin x}{x}\). We can introduce required expressions in both numerator and denominator to fit the theorem: $$\frac{\lim_{x \rightarrow 0} \frac{\sin a x}{a x}}{\lim_{x \rightarrow 0} \frac{\sin b x}{b x}}$$

Apply the theorem

Now, using the theorem, we can find the limits of both the numerator and denominator: $$\frac{\lim_{x \rightarrow 0} \frac{\sin a x}{a x}}{\lim_{x \rightarrow 0} \frac{\sin b x}{b x}} = \frac{1}{1} = 1$$

Conclusion

Therefore, by using Theorem 3.10, we found that the limit of the given function is: $$\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x} = 1$$

Key concepts

L'Hôpital's Rule

When tackling complex limit problems in calculus, L'Hôpital's Rule is a powerful tool for determining the limit of a fraction where both numerator and denominator approach zero (0/0) or infinity (∞/∞) as the variable approaches a particular value. To apply L'Hôpital's Rule, we differentiate the numerator and denominator separately and then take the limit of the new fraction. For the problem at hand, we don't need L'Hôpital's Rule since the original limits can be solved using trigonometric limit properties, but it's good to know as an alternative approach if the limits evaluation does not readily simplify.

L'Hôpital's Rule is particularly useful in cases where direct substitution leads to an indeterminate form. If the subsequent limits result in another indeterminate form, we may repeatedly apply L'Hôpital's Rule until a determinate value can be found. However, always ensure that the functions are differentiable before applying L'Hôpital's Rule.

Trigonometric Limits

Calculating trigonometric limits is an essential skill in calculus, especially when dealing with functions that involve sine, cosine, and other trigonometric functions as they approach a particular point. A well-known and frequently used trigonometric limit is \(\frac{\text{sin}(x)}{x}\) as \(x\) approaches 0, which equals 1. This property plays a crucial role in evaluating limits similar to the given exercise.

The key to solving trigonometric limits often lies in algebraic manipulation to transform the given limit into a form where this property, or similar ones, can be applied. It's also important to be mindful of the domain of trigonometric functions to avoid undefined expressions. Other trigonometric limits, such as those involving tangent or secant functions, might require additional strategies or identities for simplification.

Limit Properties

Understanding limit properties is foundational in computing the limits of functions. These properties are rules that describe how limits behave under certain mathematical operations, such as addition, subtraction, multiplication, division, and composition of functions. For example, the property that the limit of a quotient is equal to the quotient of the limits (if the limits exist and the denominator does not approach zero) is utilized in the exercise.

More formally, if \( \text{lim}_{x\to c} f(x) = L \) and \( \text{lim}_{x\to c} g(x) = M \), and both L and M are finite numbers, then:\( \text{lim}_{x\to c} [f(x) / g(x)] = L / M \), provided that M is not zero. It is crucial in calculus to not only use these properties but also understand when they can be applied. Incorrect application can lead to erroneous solutions, so checks such as ensuring the individual limits exist separately are important.

Sine Function Behavior

The sine function behavior is particularly smooth and predictable near the origin, which is leveraged in solving trigonometric limit problems like the one given. The sine function has a few key characteristics that are helpful in calculus:

• Sine is continuous everywhere and differentiable everywhere.
• It is periodic with a period of \(2\text{ }\text{π}\).
• Near \(x = 0\), the graph of sine function very closely resembles the line \(y = x\), which leads to the limit \(\frac{\text{sin}(x)}{x} = 1\) as \(x\) approaches 0.

In the exercise provided, by understanding the sine function behavior, we can intuitively grasp why the limit of \(\frac{\text{sin}(a x)}{a x}\) equals 1 as \(x\) approaches 0. The sine function's reliability around the origin proves useful in many areas of calculus, including series expansion and approximation of functions.