Question

Use a graphing utility to plot \(y=\frac{\sin p x}{\sin q x}\) for at least three different pairs of nonzero constants \(p\) and \(q\) of your choice. Estimate \(\lim _{x \rightarrow 0} \frac{\sin p x}{\sin q x}\) in each case. Then use your work to make a conjecture about the value of \(\lim _{x \rightarrow 0} \frac{\sin p x}{\sin q x}\) for any nonzero values of \(p\) and \(q\)

Short answer

Answer: Using our conjecture from the solution, for nonzero constants p = 8 and q = 10, we expect that the limit as x approaches 0 would be approximately equal to \(\frac{8}{10} = \frac{4}{5}\). Therefore, the expected value of the limit is \(\frac{4}{5}\).

Step-by-step solution

Choose three different pairs of nonzero constants p and q

We choose the following three pairs of values for p and q: - Pair 1: p = 2, q = 3 - Pair 2: p = 4, q = 5 - Pair 3: p = 6, q = 7

Plot the functions using a graphing utility

Using a graphing utility (e.g. Desmos, GeoGebra, or WolframAlpha), plot the following three functions: 1. \(y=\frac{\sin(2x)}{\sin(3x)}\) 2. \(y=\frac{\sin(4x)}{\sin(5x)}\) 3. \(y=\frac{\sin(6x)}{\sin(7x)}\) Make sure to zoom in near the origin (x = 0) to better understand the behavior of the function as x approaches 0.

Estimate the limit as x approaches 0

Looking at the graphs of the three functions as x approaches 0, we can make these estimations: 1. For pair 1 (p = 2, q = 3): \(\lim_{x \rightarrow 0} \frac{\sin 2x}{\sin 3x} \approx \frac{2}{3}\) 2. For pair 2 (p = 4, q = 5): \(\lim_{x \rightarrow 0} \frac{\sin 4x}{\sin 5x} \approx \frac{4}{5}\) 3. For pair 3 (p = 6, q = 7): \(\lim_{x \rightarrow 0} \frac{\sin 6x}{\sin 7x} \approx \frac{6}{7}\)

Make a conjecture about the limit for any nonzero values of p and q

Based on the observed behavior of the function and the estimations from the previous step, we can make the following conjecture: For any nonzero values of p and q, the value of \(\lim_{x\rightarrow 0} \frac{\sin px}{\sin qx}\) is approximately equal to \(\frac{p}{q}\). So, for general nonzero p and q, we can express the limit as: $$\lim_{x \rightarrow 0} \frac{\sin px}{\sin qx} \approx \frac{p}{q}$$

Key concepts

Graphing Utility

To understand how a graphing utility functions, let's dive into its basic features. A graphing utility is a software tool that allows users to plot mathematical equations and functions. These utilities are designed to visually represent functions for better interpretation and analysis. Common graphing utilities include tools like Desmos, GeoGebra, and WolframAlpha.

These tools are extremely helpful in calculus and algebra, especially when evaluating limits and analyzing behaviors of functions. Using a graphing utility, you can input functions such as \(y=\frac{\sin(px)}{\sin(qx)}\) and observe how the graph behaves. For this exercise, you would plot the function for different values of \(p\) and \(q\) to estimate the limit as \(x\) approaches zero.

When using a graphing utility:

• Input your chosen functions.
• Zoom in and out to observe behavior near particular values, such as near the origin for this exercise.
• Make observations based on the plotted graphs.

These observations facilitate estimating limits, and identifying patterns. This becomes clear when visualizing the function's behavior and testing conjectures.

Sine Function

Recognizing the sine function, \(\sin(x)\), is crucial in trigonometry. It's a periodic function that oscillates between -1 and 1. This property is leveraged in many mathematical applications, particularly in limit problems.

In the context of limits, the sine function's notable characteristic is how it behaves as its argument approaches zero. For small values of \(x\), \(\sin(x)\) is approximately equal to \(x\). This approximation stems from the Taylor series expansion of the sine function, which is \(\sin(x) \approx x - \frac{x^3}{6} + \cdots\). So, as \(x \to 0\), \(\sin(x) \approx x\).

When working with expressions like \(\frac{\sin(px)}{\sin(qx)}\), it's beneficial to note:

• \(\sin(px) \approx px\) for small \(x\).
• \(\sin(qx) \approx qx\) for small \(x\).

Thus, the expression \(\frac{\sin(px)}{\sin(qx)}\) simplifies to \(\frac{px}{qx} = \frac{p}{q}\) as \(x \to 0\). This is why the conjecture made in the exercise holds true.

L'Hôpital's Rule

L'Hôpital's Rule is a method for finding limits of indeterminate forms. It is used when direct substitution in a limit problem results in an indeterminate form like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). This rule states that if the limit of functions \(f(x)\) and \(g(x)\) take an indeterminate form at a certain point, then:

\[ \lim_{{x \to c}} \frac{f(x)}{g(x)} = \lim_{{x \to c}} \frac{f'(x)}{g'(x)} \]

if the limit on the right side exists. Here, \(f'(x)\) and \(g'(x)\) are the derivatives of \(f\) and \(g\), respectively.

• This rule simplifies the evaluation of limits that are initially unclear.
• It's particularly useful for rational functions, and those involving sine and cosine.

In the given exercise, understanding L'Hôpital's Rule can affirm the conjecture made. For instance, when \(x \to 0\), \(\frac{\sin(px)}{\sin(qx)}\) leads to the indeterminate form \(\frac{0}{0}\). Applying the rule once, we derive \( \frac{p \cos(px)}{q \cos(qx)}\), which simplifies to \(\frac{p}{q}\) as \(x \to 0\). However, since the limit was already conjectured based on the pattern observed graphically, using this rule serves as a verification method.