Problem 21 - Precalculus with Limits

Question

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to 0} \dfrac{\sin^2\ x}{x}$$

Step-by-Step Solution

Answer

The limit of \(\dfrac{\sin^2 x}{x}\) as \(x\) approaches zero can be estimated numerically by observing the trend in the function values for values of \(x\) close to zero, and confirmed graphically by inspecting the trend in the function's graph near \(x=0\). While the exact numerical value requires more advanced mathematical techniques or tools, the process outlined here offers a good estimation method.

Step 1: Set Up the Table

Choose values of \(x\) that approach zero from both the positive and negative sides and calculate the corresponding function values. For example, you could select values like -0.1, -0.01, -0.001, 0.001, 0.01, 0.1.

Step 2: Calculate Function Values

Substitute each chosen value of \(x\) into the function \(\dfrac{\sin^2 x}{x}\) and calculate the corresponding function value. Store these results in the table.

Step 3: Observe the Trend

As \(x\) approaches 0, observe the trend in the function values. As the values of \(x\) get closer and closer to zero, the function values should approach a certain number, which will be the estimated limit.

Step 4: Graph the Function

Plot the function \(\dfrac{\sin^2 x}{x}\) using a graphing utility. The limit as \(x\) approaches 0 should be visible on the graph. As \(x\) gets close to zero, the function should approach the estimated limit. The graphical result should confirm the numerical estimation.

Key Concepts

Numerical Estimation

Numerical estimation is a practical method to find the limit of a function. It's especially useful when dealing with functions that are complex or difficult to solve analytically. To estimate the limit numerically, you create a table of values for the function as it approaches a specific point. In this exercise, we explore the behavior of the function \( f(x) = \frac{\sin^2 x}{x} \) to find the limit as \( x \to 0 \).
The process begins by selecting values for \( x \) that are increasingly close to zero, both from the left (negative) and right (positive) sides.

• For example, use \( x = -0.1, -0.01, -0.001, 0.001, 0.01, 0.1 \).
• Substitute these values into the function to compute the output values.

As you construct the table, you will notice a trend in the function values. When you observe this trend, it will help you identify the limit. The values of the function \( \frac{\sin^2 x}{x} \) should converge to a specific number as \( x \) gets closer to zero, thereby giving an estimation of the limit.

Graphing Utility

Graphing utilities are powerful tools that visually represent mathematical functions. They are particularly useful for verifying results obtained through numerical estimation. In the exercise, you use a graphing utility to confirm the limit of \( \frac{\sin^2 x}{x} \) as \( x \to 0 \).
Once you have calculated the function values and estimated the limit, graph the function using the graphing utility. This software allows you to:

• Plot the function across a chosen range. A typical range would include values very close to the target point—in this case, zero.
• Zoom in on the graph to see how the function behaves as \( x \) approaches zero.
• Visually confirm the asymptotic behavior, observing that the function appears to approach a particular value.

This graphical approach not only confirms your numerical estimation but also enhances understanding by providing a visual perspective on the behavior of the function near the point of interest.

Trigonometric Functions

Trigonometric functions, like sine and cosine, play a crucial role in calculus and throughout mathematics. Functions that involve these trigonometric components often require special consideration, especially when dealing with limits. In this exercise, the function \( f(x) = \frac{\sin^2 x}{x} \) involves the sine function, and we aim to find its limit as \( x \to 0 \).
The sine function has distinct properties as \( x \) approaches zero, one of which is that \( \sin x \approx x \) when \( x \) is near zero.
To appropriately manage trigonometric functions in limits:

• Recognize common small-angle approximations, such as \( \sin x \approx x \), which are helpful in estimating limits around zero.
• Consider rewriting or manipulating the function using identities or approximations if direct substitution is complex.
• Understand how these functions are continuous and how their limits play out visually on graphs.

Grasping the behavior of trigonometric functions near zero can demystify the process of limit estimation and make challenging problems more tractable.