Problem 63 - Precalculus with Limits

Question

In Exercises 63-68, use a graphing utility to graph the function and the equations \(y = x\) and \(y = -x\) in the same viewing window. Use the graph to find \( \lim_{x \to 0} f(x) \). \(f(x) = x\ \cos\ x\)

Step-by-Step Solution

Answer

The limit of \(f(x) = x\ cos\ x\) as \(x\) approaches 0 is 0.

Step 1: Graph the functions using a graphing tool

Use a graphing tool to plot the three functions. \(f(x)= x\ cos\ x\), \(y = x\) and \(y = -x\) in the same viewing window. Observe their intersection points.

Step 2: Observe behavior of functions as x approaches 0

Examine the graph. Pay special attention to the behavior of the functions near \(x = 0\). Specifically, observe the behavior of \(f(x)\) as \(x\) approaches zero from the left and from the right.

Step 3: Find the limit of f(x) as x approaches 0

From the graph, determine the limit of \(f(x)\) as \(x\) approaches 0. If \(f(x) = L\) as \(x\) approaches 0, this means the limit of \(f(x)\) as \(x\) approaches 0 is \(L\).

Key Concepts

Graphing Functions

Graphing functions is a fundamental skill in understanding how equations behave visually. When graphing a function, consider the entire equation and its components. This often involves identifying key points like intersections, maxima, or minima. In our task, we were asked to graph the function \( f(x) = x\cos x \) along with the lines \( y = x \) and \( y = -x \). Observing these graphs together helps us understand the interaction between the functions as they share the same viewing window. A graph provides a visual representation of mathematical behavior. It helps identify points where graphs intersect or determine trends as \( x \) approaches certain values. In particular, focus on how close each function comes to intersecting the X or Y-axis, since these details are crucial in assessing limit behavior or where equations may overlap.Using graphing software offers additional insights, such as zooming into areas of interest—for instance, around \( x = 0 \)—to examine how closely the curves converge or diverge. Graphing is a dynamic tool that brings abstract concepts to life, allowing us to visually verify calculations and predictions.

Trigonometric Functions

Trigonometric functions, like \(\cos x\), have distinctive properties that can influence the shape of a graph. They are periodic, which means they repeat values in regular intervals. The cosine function specifically oscillates between -1 and 1 over a period of \(2\pi\). This oscillation is critical when multiplied by other factors, such as \(x\) in our example, \(f(x) = x \cos x\).Multiplying \(\cos x\) by \(x\) alters its usual periodic pattern:

• The amplitude, or height, of the oscillation changes as \(x\) changes.
• Near \(x = 0\), the product \(x \cos x\) impacts the graph significantly due to \(x\) approaching zero, reducing the function's magnitude at that point.
• As \(x\) grows, so does the stretch of the cosine wave away from the axis.

Understanding the behavior of trigonometric functions helps in predicting how changes in an equation, like adding a multiplier, alter the graph's appearance. Viewed as part of a graph, the cosine component drives the oscillatory nature, while the \(x\) scaling controls the wave's stretch and amplitude relative to the axis.

Limit Evaluation

Limit evaluation is essential in calculating how functions behave as \(x\) becomes very close to a specific point, often zero. In this exercise, the focus is on evaluating the limit of \(f(x) = x\cos x\) as \(x\) approaches 0.Limits denote the value a function heads toward as \(x\) comes infinitesimally close to the target point. Observing the graph of \(f(x)\), we zoom in on \(x = 0\) to see:

• How the values of \(f(x)\) change as \(x\) moves closer to zero from both negative and positive sides.
• Whether the function tends toward the same value from both directions, indicating a single, definite limit.

For \( f(x) = x\cos x \), the function simplifies when \( x \to 0 \). Since \( \cos(0) = 1 \), we multiply it by 0, leading to a limit of zero. Evaluating limits graphically helps verify analytical solutions and understand behavior changes around critical points in the function's domain.Using both graphical and analytical methods allows a full comprehension of the function's behavior, ensuring that even subtle changes around limits are accurately captured.