Problem 42 - Precalculus with Limits

Question

In Exercises 37-48, use a graphing utility to graph the function and approximate the limit accurate to three decimal places. $$\lim_{x \to 0} \dfrac{\sin\ 3x}{x}$$

Step-by-Step Solution

Answer

As per the graphical observation, the limit as \(x\) approaches 0 for the function \(\dfrac{\sin \ 3x}{x}\) is approximately 3.

Step 1: Utilize the Graphing Utility

Set up the function \(\dfrac{\sin\ 3x}{x}\) in your graphing utility. This function will exhibit a discontinuity at \(x=0\). However, it is the behaviour of the function around this point that is most relevant for calculating the limit.

Step 2: Zoom In Near the Point \(x=0\)

Using the graph utilities, zoom in near the point \(x=0\). This will allow for a closer examination of the behaviour of the function as \(x\) approaches 0. Note how the function behaves as \(x\) gets closer and closer to 0 from both left and right.

Step 3: Estimate the limit from the graph

Leverage the graphical tool to estimate the limit of the function as \(x\) approaches 0. It is crucial to get this accurate to three decimal places. Most of the time, graphing utilities also provide an option to evaluate limits numerically, in case visual estimate is not accurate.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool that can help visualize mathematical functions. It's particularly helpful for understanding limits as it provides a graphical representation of how a function behaves near a particular point. When dealing with complex functions, like \(\lim_{x \to 0} \dfrac{\sin\ 3x}{x}\), the graphing utility allows you to input the function and see its graph instantly.

The key benefits of using a graphing utility include:

• Quickly graphing functions to visualize their behavior.
• Zooming into specific areas of a graph to observe how a function behaves as it approaches a certain value.
• Automatically calculating limits and other numerical properties of the function.
  
  

While setting up the function \(\dfrac{\sin\ 3x}{x}\), you can pay close attention to how the graph behaves around \(x=0\). This specific point is crucial as the limit examines the function's behavior as \(x\) approaches 0.

Discontinuity

Discontinuity in a function occurs when there is a sudden jump or hole at a particular point. In our function \(\dfrac{\sin\ 3x}{x}\), there's a noticeable discontinuity at \(x=0\). However, understanding this doesn't have to be too tricky!

Even though the function cannot be evaluated exactly at \(x=0\), we can explore the function's behavior very close to this point. A function can still exhibit a trend or direction as \(x\) gets closer to a point of discontinuity.

• Steps like zooming in near the point of discontinuity can help understand how the function behaves before and after the point.
• In cases where functions approach a value smoothly from both sides of a discontinuity, a limit might exist despite the discontinuity itself.

A graphing utility is superior here, helping us see if the points around \(x=0\) seem to converge to a similar y-value.

Sine Function

The sine function, denoted as \(\sin\), is a fundamental trigonometric function that describes a smooth wave-like pattern. When looking at \(\dfrac{\sin\ 3x}{x}\), you're essentially examining how the sine function behaves when multiplied by the factor 3 inside its argument.

Here's what to note about the sine component:

• The multiplication by 3 affects the frequency of the sine wave, causing it to oscillate faster.
• Sine is a continuous function, but when divided by \(x\), it introduces potential discontinuity when \(x=0\).

Understanding the sine function's properties helps explain why, as \(x\) approaches 0, the overall function tends to behave smoothly. This gives insight into why the limit \(\lim_{x \to 0} \dfrac{\sin\ 3x}{x}\) might exist even with the apparent discontinuity at the denominator. By examining the graph or using algebraic techniques, one finds that this limit equates to 3, evidencing the role of sine and its transformations in the function's behavior.