Problem 19 - 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. $$\lim_{x \to 0} \dfrac{\sin\ x}{x}$$

Step-by-Step Solution

Answer

The limit of the function \(\frac{\sin x}{x}\) as \(x\) approaches 0 is 1, as confirmed both numerically and graphically.

Step 1: Identify the Function

Identify the function for which we want to compute the limit. Here, the function is \(f(x)=\frac{\sin x}{x}\).

Step 2: Create a Table of Values

Create a table of function values for \(x\) nearing 0 from two directions - left and right. Select a series of increasingly close values to 0 and compute the function value. Here, you could use the values: \(x= -0.1, -0.01, -0.001, 0.001, 0.01, 0.1\)

Step 3: Compute the Function Values

Substitute these values into the function and record the resulting function values in your table.

Step 4: Estimate the Limit

Looking at the table, you will notice that as \(x\) gets closer to 0, the function value approaches a certain number. This is the limit of the function as \(x\) approaches 0.

Step 5: Confirm Graphically

Plot the function using a graphing tool. Observe the behavior of the function as \(x\) approaches 0. The function should approach the same value as calculated in Step 4. This will confirm your result graphically.

Key Concepts

Numerical Estimation

When dealing with limits, numerical estimation offers a way to approach understanding the behavior of a function as it nears a certain point. This can be particularly useful when the limit is not immediately obvious. To estimate numerically, one generally selects a series of values that get closer and closer to the point of interest—in our case, 0. By evaluating the function at these points, a pattern often emerges.

• Choose points that lie very close to the target point (for instance, -0.1, -0.01, -0.001, 0.001, 0.01, 0.1).
• Calculate the function value for each point.
• Observe how these values change as they near the target point.

The clearer this pattern becomes, the more we can trust our numerical estimate of the limit. It serves as a way to visualize the nearing behavior of the function.

Graphing Utility

Graphing utilities are indispensable tools for confirming numerical estimations visually. These digital resources can plot complex functions easily, allowing students to see the limiting behavior as the variable approaches specific values. In this context, the graphing tool will let you observe the function \(rac{\sin x}{x}\) as x approaches 0.Here's how you can use a graphing utility effectively:

• Input the function into the graphing tool.
• Set the scale to focus on values near the point of interest.
• Observe the curve as it closes to the point, noticing its trend.

This graphic representation should confirm the numerical estimations, showcasing how the function behaves in real-time as x becomes closer to 0, helping to solidify the concept of the limit.

Sine Function

The sine function, denoted as \(\sin x\), is a fundamental trigonometric function representing periodic oscillations. It is central to many mathematical and real-world applications. In the context of limits, \(\sin x\) behaves interestingly, especially when divided by x as in our function \(\frac{\sin x}{x}\).Key features of the sine function to remember:

• It oscillates between -1 and 1.
• It is an odd function, meaning \(\sin(-x) = -\sin(x)\).
• Its value at 0 is \(\sin(0) = 0\).

Understanding these characteristics aids in comprehending why \(\frac{\sin x}{x}\) approaches a definite value as x approaches 0.

Calculating Limits Numerically

Calculating limits numerically involves using both numerical estimation and graphical verification. By explicitly analyzing function behaviors at select points, students can predict a limit accurately. Using the function \(\frac{\sin x}{x}\), this involves a few intuitive steps:

• Compute function values at points increasingly near zero.
• Record results systematically, noting any trends towards a specific numerical value.

Once these steps are completed numerically, employ a graphing utility to ensure the results align with visual trends.As x approaches 0, you should notice that the function approaches a value of 1—this is the limit mathematically and numerically confirmed through estimation. This type of analysis not only provides an answer, but it deepens understanding by combining computational work with visual evidence.