Question

In Exercises 7–14, create a table of values for the function and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 0} \frac{\sin 2 x}{x} $$

Short answer

Through the table and the graph, we can estimate that the limit as \( x \) approaches 0 of \( \frac{\sin 2x}{x} \) is 2.

Step-by-step solution

Create Table

Let's create a table to compute the values of \( \frac{\sin 2x}{x} \) at points increasingly close to 0. This will help us estimate the limit. We will compute the function value for both positive and negative values of x.

Compute Function Values

Now we compute the function's values at \( x = -0.1, -0.01, 0, 0.01, 0.1 \) to see if there is a pattern or a number that the function is getting closer to as \( x \) approaches 0.

Estimate Limit

By inspecting the results from the table, a pattern should emerge that will give us an estimate for the limit as \( x \) approaches 0. The function seems to be getting closer to a certain number.

Confirm with Graph

Using a graphing utility, graph the function \( \frac{\sin 2x}{x} \) to visually confirm the estimate. The limit is where the function value is tending as \( x \) approaches 0.

Key concepts

Limits in Calculus

In calculus, the concept of a limit is fundamental to understanding how functions behave as they approach a particular value. To say that the limit of a function as the variable approaches a certain value is to describe the tendency of the function's output as the input gets closer to that value. It's a way of analyzing the immediate surroundings of a point, even if the function is undefined exactly at that point.

For example, in the exercise, we are concerned with the limit of the trigonometric function \( \frac{\sin 2x}{x} \) as \( x \) approaches zero. Finding this limit helps us explore the behavior of this function near \( x = 0 \) and is essential in fields like physics and engineering, where such expressions often emerge.

Trigonometric Limits

Trigonometric limits are special cases in calculus involving trigonometric functions such as sine, cosine, and tangent. These functions can oscillate and behave in unique ways near certain points, which is why their limits require careful consideration. A common strategy to deal with trigonometric limits is to use known limit forms, such as the standard \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \).

The provided exercise involves a similar limit, and understanding these predefined limits can greatly simplify the process. In more complex cases, trigonometric identities might also be used to manipulate the expression into a more familiar form before evaluating the limit.

Graphing Utility

A graphing utility is an incredibly helpful visual tool in calculus that allows students to comprehend the behavior of functions graphically. By plotting the function over a range of values, one can visually inspect and confirm the behavior of the limit that was estimated algebraically or numerically. In the exercise, confirming the limit of \( \frac{\sin 2x}{x} \) as \( x \) approaches zero can be more intuitively done after looking at the graph.

Seeing the function flatten out or approach a line as \( x \) approaches the point of interest provides a powerful confirmation of what the number-crunching suggested. This kind of multi-modal reinforcement is excellent for deepening understanding and retention.

Limit Approximation Methods

There are several approximation methods to estimate limits when an algebraic solution is not straightforward. One such method is creating a table of values, which involves selecting input values close to the point of interest and observing the output behavior as demonstrated in our exercise. This numerical approach allows students to hypothesize the limit based on tangible results.

Another method is graphical approximation, where a function's graph presents a visual cue to the limit. Finally, algebraic manipulation, such as factoring, rationalizing, or using conjugates, can simplify an expression, making the limit more apparent. Each of these methods plays a distinct role in painting a comprehensive picture of what a function is doing as it approaches a certain point, and together, they solidify the understanding of limits, a cornerstone of calculus.