Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Sketch the graph of the function \(f\) and evaluate \(\lim _{x \rightarrow a} f(x)\), if it exists, for the given value of \(a\). \(f(x)=\left\\{\begin{array}{ll}|x| & \text { if } x \neq 0 \\ 1 & \text { if } x=0\end{array} \quad(a=0)\right.\)

Step-by-Step Solution

Answer

The graph of the function \(f(x) = \begin{cases} |x| & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}\) is a V-shaped graph following the absolute value function, with a "hole" at the origin and a "dot" at the point (0,1). Evaluating the limit as x approaches 0, we find that both one-sided limits are equal to 0. Therefore, \(\lim_{x \rightarrow 0} f(x) = 0\).

Step 1: Analyze the function for x≠0

For x≠0, the function is defined as f(x) = |x|. This is the absolute value function, which is a V-shaped graph with a vertex at the origin (0,0). It is continuous for all x-values and has values of f(x) approaching 0 as x approaches 0 from both the left and the right.

Step 2: Analyze the function for x=0

For x=0, the function is defined as f(x) = 1. So, at the point x=0, instead of following the expected V-shape with point (0,0), the function has a defined value of 1. This creates a "hole" in the graph at the origin and a "dot" at point (0,1).

Step 3: Sketch the graph

To sketch the graph, we combine the observations from Steps 1 and 2. Draw the V-shape of the absolute value function with a "hole" at the origin. Then, add a "dot" at the point (0,1) to represent the value of the function when x=0.

Step 4: Evaluate the limit as x approaches 0

Now we will evaluate the limit as x approaches 0. From the graph, we can see that as x approaches 0 from the left and right, the function f(x) approaches 0 because of the absolute value part. But, since f(x) is defined as being equal to 1 at the point x=0, we have a jump discontinuity. In order for the limit to exist, the function must approach the same value from both the left and right. In this case, we can see that: \(\lim_{x \rightarrow 0^{-}} f(x) = 0\) \(\lim_{x \rightarrow 0^{+}} f(x) = 0\) Since both one-sided limits are equal to 0, the limit exists and is equal to 0: \(\lim_{x \rightarrow 0} f(x) = 0\)

Key Concepts

Absolute Value Function

The absolute value function is represented by the notation \( |x| \) and is pivotal in understanding various mathematical concepts. It converts all negative inputs into their positive counterparts while leaving positive inputs unchanged, rendering the output of \( |x| \) always non-negative.

Visually, the graph of an absolute value function is a V-shape, symmetrically rising from the origin to both the left and the right. This symmetry results from the fact that for every value \( x \) on the number line, the points \( (x, |x|) \) and \( (-x, |-x|) \) are mirrored across the y-axis. Here's an important thing to note: although the concept might sound simple, its implications are significant in calculus, especially when dealing with limits and continuity.

Students often encounter the absolute value when addressing distance and real-world problem-solving, such as finding the shortest path or determining a difference in values without considering direction.

Graph Sketching

Sketching the graph of a function is a fundamental skill in calculus, serving not just to visualize the function but also to analyze its behavior at various points. When sketching, focus on identifying key features: intercepts, slopes, maxima, minima, and points of discontinuity.

In our given problem, we're dealing with a special case where the function changes definition at \( x=0 \). For \( x eq 0 \) the graph follows the standard V-shape of the absolute value function. At \( x=0 \) we have a different value which creates a unique point, needing to be marked distinctly on the graph – often with a dot or a circle to indicate its special status.

Quick Sketching Tips:

• Identify the overall shape (like the V-shape for the absolute value function).
• Mark key points, including where the function changes definition.
• Consider behavior at and near discontinuities carefully.

Mastering graph sketching is more than just making accurate drawings; it's about understanding the function's nature deeply enough to anticipate its behavior in different scenarios.

Discontinuity

In calculus, a discontinuity is a point at which a function is not continuous. Discontinuities can appear in various forms such as jump, point, infinite, or removable discontinuities.

A function is continuous at a point if it meets three conditions: The function is defined at that point, the limit exists, and the function's value equals the limit. Conversely, if any of these conditions fail, we have a discontinuity.

In the given problem, the function displays a jump discontinuity at \( x=0 \). Despite the V-shape of the absolute value indicating the function approaches zero from either side, the function's definition abruptly changes the value to 1 at \( x=0 \).

Understanding Limits and Discontinuity:

The limit of a function as \( x \) approaches a certain value is all about the function's behavior as \( x \) gets infinitely close to that value - not the function's actual value at the point. This distinction plays a crucial role in differentiating between the limit of a function at a point and its continuity.