Problem 20 - Precalculus: Mathematical for Calculus

Question

Use a graphing device to determine whether the limit exists. If the limit exists, estimate its value to two decimal places. $$\lim _{x \rightarrow 0} \frac{x^{2}}{\cos 5 x-\cos 4 x}$$

Step-by-Step Solution

Answer

The limit exists and is approximately 0.22.

Step 1: Understanding the Function

The expression given is \( \lim_{x \to 0} \frac{x^2}{\cos 5x - \cos 4x} \). We are required to find this limit as \( x \) approaches \( 0 \). Initially, we observe that both the numerator and denominator approach zero as \( x \) approaches \( 0 \). This suggests using a graph to better understand the behavior of this function.

Step 2: Graph the Function

Use a graphing calculator or software to graph the function \( y = \frac{x^2}{\cos 5x - \cos 4x} \) in the interval close to \( x = 0 \). This visual representation will help identify any trends or common patterns as \( x \) approaches zero, aiding in the estimation of the limit.

Step 3: Observe Graphical Behavior

Upon graphing, monitor how the function behaves as \( x \) nears \( 0 \). The graph should reveal whether the value of the function approaches a specific value or diverges. Focus on observing the pattern of the y-values on both sides of \( x = 0 \).

Step 4: Estimating the Limit

From the graph, you'll notice that as \( x \) gets very close to \( 0 \), the function's y-value approaches a particular number. Estimate this number to two decimal places. In this case, the graph should show that the limit approaches \( 0.22 \).

Step 5: Conclusion

Based on the observations from the graph, conclude that the limit exists because the function tends towards a specific value as \( x \) approaches \( 0 \). Therefore, the estimated limit is \( 0.22 \).

Key Concepts

Graphical Analysis

When approaching limit problems, especially those involving complex functions, graphical analysis becomes crucial. Visualizing a function through a graph helps us understand how it behaves when certain variables approach specific values. In the context of this problem, we're analyzing the limit as \(x\) nears 0 for the function \(y = \frac{x^2}{\cos 5x - \cos 4x}\). Graphing this function allows us to see where the y-values stabilize, which gives a visual cue on whether the limit exists.

• By using a graphing calculator or software, plot the function in a small range around \(0\).
• Pay attention to how the curve behaves on both sides of \(x = 0\).

By observing the graph, you can determine whether the y-values converge to a specific point, which indicates that the limit exists. In our case, you would notice that the y-values approach 0.22 as \(x\) gets closer to 0, thus confirming the existence of the limit.

Precalculus

Precalculus provides essential tools for analyzing and understanding limits, an important concept in calculus. Limits explore the behavior of a function as it approaches a specific point. In this exercise, we see that the expression \(\frac{x^2}{\cos 5x - \cos 4x}\) results in an indeterminate form \(\frac{0}{0}\) as \(x\) approaches 0. Precalculus helps us identify these forms and suggests methods, such as graphical analysis, to resolve them.
Understanding the foundational behaviors of functions, how they change in response to variables, and how functions like trigonometric functions behave are all part of precalculus. These insights set the stage for determining limits by providing background into:

• The continuity and discontinuity of functions.
• How closely functions approximate specific y-values as x-values near certain points.

In this problem, knowing precalculus principles allowed us to identify the indeterminate nature and use graphical methods to find the limit.

Trigonometric Functions

Trigonometric functions like cosine take a central role in this exercise. Here, we explore \(\cos 5x\) and \(\cos 4x\), which are periodic and oscillate between -1 and 1. This property impacts how \(\cos 5x - \cos 4x\) behaves as \(x\) approaches 0. As both functions approach a small angle, their differences can result in very small values, creating the indeterminate form we see at \(x = 0\).
Understanding the properties of these trigonometric functions is crucial:

• Both \(\cos 5x\) and \(\cos 4x\) are smooth and continuous around zero, meaning they don't jump or break.
• They have nearly identical outputs for values of \(x\) close to 0, which can lead to very small differences—hence the denominator nearly reaches zero.

Recognizing these properties aids in predicting how the function behaves, especially around critical points like \(x = 0\). In limit problems, especially those involving such close oscillations, graphical interpretation often supports analytical methods to find the limit.