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: The conjectured value of the limit as x approaches 0 for any non-zero values of p and q is given by the equation: \(\lim_{x \rightarrow 0} \frac{\sin p x}{\sin q x} = \frac{p}{q}\)

Step-by-step solution

Select three pairs of non-zero constant values for p and q

Choose three pairs of non-zero constant values. For this example, let's use the following pairs: - \((p_1, q_1) = (1, 2)\) - \((p_2, q_2) = (3, 4)\) - \((p_3, q_3) = (5, 6)\)

Plot the function for each pair of constants

Using a graphing utility, plot the given function \(y=\frac{\sin p x}{\sin q x}\) for each pair of constants \((p, q)\). You should obtain three different graphs.

Estimate the limit as x approaches 0 for each case

For each graph, estimate the value of \(\lim_{x \rightarrow 0} \frac{\sin p x}{\sin q x}\) by looking at the graph near \(x=0\). For example, in this step, we would calculate the limit: - For \((p_1, q_1) = (1, 2)\): \(\lim_{x \rightarrow 0} \frac{\sin 1x}{\sin 2x}\) - For \((p_2, q_2) = (3, 4)\): \(\lim_{x \rightarrow 0} \frac{\sin 3x}{\sin 4x}\) - For \((p_3, q_3) = (5, 6)\): \(\lim_{x \rightarrow 0} \frac{\sin 5x}{\sin 6x}\)

Make a conjecture about the general case for the limit as x approaches 0

Based on your observations, propose a conjecture about the value of \(\lim_{x \rightarrow 0} \frac{\sin p x}{\sin q x}\) for any non-zero values of \(p\) and \(q\). In this case, you could conjecture that the limit as \(x\) approaches 0 for any values \((p, q)\) can be represented by the equation: \(\lim_{x \rightarrow 0} \frac{\sin p x}{\sin q x} = \frac{p}{q}\)

Key concepts

Trigonometric functions

Trigonometric functions, like sine and cosine, are fundamental in understanding periodic phenomena. The sine function, written as \( \sin(x) \), is especially significant because it describes how a point on a unit circle oscillates. Here are some key properties of the sine function:

• The function's range is from -1 to 1.
• It is periodic with a period of \(2\pi\).
• It is odd, meaning \(\sin(-x) = -\sin(x)\).

When dealing with expressions like \(\sin(p x)\), the parameters \(p\) and \(q\) that modify the angle affect the frequency and amplitude of the oscillation. These modifications play a crucial role in understanding limits at zero, as they can alter how the function behaves when \(x\) approaches zero.

Graphing utility

Graphing utilities are powerful tools that allow us to visualize mathematical functions easily. By inputting different mathematical expressions into a graphing calculator or software, you can generate graphs that provide insights unattainable by mere calculation.When plotting \( y = \frac{\sin(px)}{\sin(qx)} \), you can observe how the graph behaves as \(x\) approaches zero. Key factors to effectively using a graphing utility include:

• Selecting an appropriate range and scale for \(x\) to accurately capture the behavior near zero.
• Zooming into regions near \(x=0\) to better estimate limits.
• Choosing diverse values for \(p\) and \(q\) to examine different scenarios.

Graphing utilities simplify the process of forming hypotheses about function behavior, such as estimating limits of functions as they approach specific values.

Conjecture

A conjecture is an educated guess or hypothesis that captures an underlying pattern observed across multiple instances or cases. In mathematics, forming a conjecture involves:

• Analyzing a variety of examples.
• Looking for common behaviors or results.
• Framing a general rule based on observed data.

In this exercise, after plotting \( y = \frac{\sin(px)}{\sin(qx)} \) for different pairs \((p, q)\) and observing the limit as \(x\) approaches zero, a conjecture is made: \[\lim_{x \rightarrow 0} \frac{\sin(px)}{\sin(qx)} = \frac{p}{q}\].This conjecture suggests a consistent relationship, observed visually and experimentally, providing a hypothesis that can be tested with further mathematical rigor or analytical methods.

Limits at zero

Understanding limits at zero is a pivotal aspect of calculus, particularly when dealing with behaviors of functions as they approach a specific point. In this exercise, we are interested in the behavior of the function \( \frac{\sin(px)}{\sin(qx)} \) as \(x\) nears zero.Here’s how limits at zero are academically useful:

• They help determine a function's behavior in close proximity to the point of interest.
• Providing insights into continuity and differentiability.
• Enabling estimation of function values in small intervals.

In situations involving ratios of sine functions, such as \( \lim_{x \rightarrow 0} \frac{\sin(px)}{\sin(qx)} \), we find that the limit reflects the coefficients \(p\) and \(q\), leading to the conclusion \( \frac{p}{q} \). This result emerges by applying L'Hôpital's Rule or using standard limit properties that simplify the behavior of trigonometric functions near zero.