Problem 41 - 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. $$\lim_{x \to 0} \dfrac{\sin\ 2x}{x}$$

Step-by-Step Solution

Answer

The limit of the function \( \frac{\sin\ 2x}{x} \) as x approaches 0, accurate to three decimal places, is approximately 2.000.

Step 1: Graphing the function

Firstly, you need to input the function \( \frac{\sin\ 2x}{x} \) in your graphing utility. You may notice that, as this is a fraction whose numerator is sin(2x) and no x in the denominator, there would be a problem when x=0 (undefined). However, this does not mean the limit does not exist.

Step 2: Identifying the limit as x approaches 0

On the graph, you will see that as x approaches to 0 from the left or the right, the y-values are getting closer to a certain value. This can be roughly estimated to be 2. Despite a hole at x=0, the function seems to be getting closer to this value as x approaches 0.

Step 3: Approximating the limit to three decimal places

Most of the graphing utilities provide the feature to evaluate y-values for given x-values. Using this feature, check for x-values close to 0 (like 0.0001 and -0.0001) and see what the y-values are. This will give an approximation of the limit. For the function in this exercise, the limit as x approaches 0, rounded to three decimal places, is approximately 2.000.

Key Concepts

Limit of sin(2x)/x as x Approaches 0

Understanding the limit of a function is a fundamental concept in calculus. Specifically, the limit of the function \( \sin(2x)/x \) as \( x \) approaches 0 can be a tricky concept because directly substituting \( x = 0 \) leads to an undefined expression, since division by zero is not allowed.

However, through L'Hôpital's Rule, which states that if the limit of functions \( f(x) \) and \( g(x) \) as \( x \) approaches a point leads to an indeterminate form \( 0/0 \), we can take the derivative of the numerator and denominator separately and then find the limit of the resulting expression. In this case, the derivatives of \( \sin(2x) \) and \( x \) are \( 2\cos(2x) \) and \( 1 \) respectively. As \( x \) approaches zero, \( \cos(2x) \) approaches \( \cos(0) \), which is 1. Therefore, the limit of \( \sin(2x)/x \) as \( x \) approaches 0 is \( 2 \cdot 1 = 2 \).

This concept is essential in understanding and handling functions that are not immediately obvious at certain points, especially when dealing with trigonometric functions.

Evaluating Limits Graphically

One can approximate the limit of a function graphically using a graphing utility. When you graph the function \( \frac{\sin(2x)}{x} \), you'd want to observe the behavior of the y-values as the x-values get closer to 0. This visual representation is helpful because it literally 'draws out' the tendency of the function as one input variable approaches a specific value.

Here's how you can proceed: Zoom in near the point where \( x \) is approaching 0, and notice the y-values. If they are stabilizing or converging to a number, that number is the approximate limit. The more you zoom in, without the graph significantly changing shape around that point, the more confident you can be about the limit. This graphical method provides a good approximation, but for a precise value, one would have to rely on calculative methods or more advanced features of the graphing utility.

Undefined Limits at a Point

In the world of calculus, not all limits are defined. When we talk about undefined limits, we refer to the behavior of a function at certain points where the function does not approach a specific value. Take the point \( x = 0 \) for the function \( \frac{\sin(2x)}{x} \), for example. If you try to substitute 0 directly into the function, the expression becomes \( \frac{\sin(0)}{0} \) which is undefined, because \( \sin(0) \) is 0, and now you have division by zero.

However, this undefined behavior at the point itself does not mean that the limit does not exist as \( x \) approaches that point. The limit instead refers to the value that the function approaches, not the value it takes at that very point. Thus, despite the function not being defined at \( x = 0 \) due to division by zero, the limit as \( x \) approaches 0 exists and as demonstrated in the previous sections, is 2. Such instances are key in highlighting the concept of limits approaching but not necessarily reaching a point on a function.