Problem 57 - Thomas Calculus

Question

Exploring (sin \(k x ) / x\) Graph \(y=(\sin x) / x, y=(\sin 2 x) / x,\) and \(y=(\sin 4 x) / x\) together over the interval \(-2 \leq x \leq 2 .\) Where does each graph appear to cross the \(y\) -axis? Do the graphs really intersect the axis? What would you expect the graphs of \(y=(\sin 5 x) / x\) and \(y=(\sin (-3 x)) / x\) to do as \(x \rightarrow 0 ?\) Why? What about the graph of \(y=(\sin k x) / x\) for other values of \(k ?\) Give reasons for your answers.

Step-by-Step Solution

Answer

For \( y = \frac{\sin(kx)}{x} \) as \( x \to 0 \), the graphs approach the y-axis values of \( k \), but do not truly intersect. This is due to the limit \( \lim_{x \to 0} \frac{\sin(kx)}{x} = k \).

Step 1: Understanding the Function Behavior

The function of interest is of the form \( y = \frac{\sin(kx)}{x} \). As \( x \to 0 \), both \( y = \frac{\sin(kx)}{x} \) and \( y = \frac{\sin(x)}{x} \) have an indeterminate form \( \frac{0}{0} \). It is useful to recall that \( \lim_{x \to 0} \frac{\sin(kx)}{x} = k \). This property will help us analyze the graphs.

Step 2: Plotting the Functions

Graphically, plot \( y = \frac{\sin(x)}{x}, \frac{\sin(2x)}{x}, \) and \( \frac{\sin(4x)}{x} \) over the interval \( -2 \leq x \leq 2 \). Notice that these functions have undefined values at \( x = 0 \), but the limits at \( x = 0 \) are determinate based on the behavior at small \( x \).

Step 3: Analyzing the y-axis Intersection

For \( y = \frac{\sin(x)}{x} \), at \( x = 0 \), the limit is \( \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \). Similarly, \( y = \frac{\sin(2x)}{x} \) gives a limit of 2, and \( y = \frac{\sin(4x)}{x} \) gives a limit of 4 as \( x \to 0 \). Hence, they "appear" to meet the y-axis at these values.

Step 4: Predicting Behavior of Other Graphs

For \( y = \frac{\sin(5x)}{x} \), as \( x \to 0 \), we expect \( \lim_{x \to 0} \frac{\sin(5x)}{x} = 5 \). Similarly, for \( y = \frac{\sin(-3x)}{x} \), the limit should be -3. Generally, for any \( k \, (y = \frac{\sin(kx)}{x}) \), as \( x \to 0 \), the value approaches \( k \), due to the nature of the sine function and its derivative.

Step 5: Conclusion on General Case

The behavior as \( x \to 0 \) can be concluded for any \( k \) that \( \lim_{x \to 0} \frac{\sin(kx)}{x} = k \). This holds because \( \sin(kx) \approx kx \) when \( x \) is very small. Thus, it approaches \( \frac{kx}{x} = k \). Hence, the graphs do not actually intersect the y-axis at a point of continuity, but they have predictable behavior approaching the y-axis.

Key Concepts

Sine function

The sine function is a fundamental trigonometric function that oscillates between -1 and 1. It is periodic with a period of \( 2\pi \), which means it repeats every \( 2\pi \) units along the x-axis. The sine function is commonly used in mathematics to model waves and oscillations, and its properties are pivotal in analyzing trigonometric limits.
For small angles, the sine of an angle \( \theta \) is approximately equal to the angle itself when measured in radians. This approximation, \( \sin(\theta) \approx \theta \) for small \( \theta \), is crucial when analyzing the behavior of the function \( \frac{\sin(kx)}{x} \) as \( x \) approaches zero.

• Behavior at Small Values: As \( x \to 0 \), \( \sin(kx) \approx kx \), simplifying the evaluation of limits.
• Limit Computation: Therefore, the limit \( \lim_{x \to 0} \frac{\sin(kx)}{x} = k \), because \( \frac{kx}{x} = k \).

Understanding the sine function helps grasp the limiting behavior of trigonometric functions as variables approach specific values.

Indeterminate form

The term "indeterminate form" often arises in calculus, especially when evaluating limits. It refers to expressions that do not initially provide enough information to determine a limit, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These require further manipulation to find a meaningful limit.
For the functions \( y = \frac{\sin(kx)}{x} \), the indeterminate form \( \frac{0}{0} \) occurs at \( x = 0 \). To handle this, we apply L'Hôpital's Rule or recognize small angle approximations. \( \lim_{x \to 0} \frac{\sin(kx)}{x} = k \) is derived by small angle approximation, where \( \sin(kx) \) closely matches \( kx \) as \( x \) becomes very close to zero.

• L'Hôpital's Rule: Can be applied when a limit results in \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
• Importance: Understanding indeterminate forms is essential for evaluating the behavior of functions at critical points.

Mastering indeterminate forms allows students to tackle complex limits with confidence.

Graphing functions

Graphing functions allows us to visually explore behavior over an interval and around specific values. For the expressions \( y = \frac{\sin(kx)}{x} \), creating visual representations on a graph is invaluable.
Over the interval \( -2 \leq x \leq 2 \), we see how these graphs "appear" to intersect the y-axis, despite having undefined values at \( x = 0 \). Their behavior as \( x \to 0 \) helps visualize limits and predict function behavior.

• Observing Oscillations: Sine functions show periodicity and smooth oscillations that can be visually tracked.
• Y-axis Appearance: The limit predictions \( \lim_{x \to 0} \frac{\sin(kx)}{x} = k \) cause graphs to meet the y-axis at the predicted limit values.
• Predicting Future Graphs: By understanding the behavior of \( \frac{\sin(kx)}{x} \), we can predict graphs for any value of \( k \).

Graphing solidifies abstract mathematical predictions into tangible observations, providing insight into functional characteristics and their limits.