Problem 50 - Calculus: Early Transcendentals

Question

A step function Let \(f(x)=\frac{|x|}{x},\) for \(x \neq 0\) a. Sketch a graph of \(f\) on the interval [-2,2] b. Does \(\lim f(x)\) exist? Explain your reasoning after first examining \(\lim _{x \rightarrow 0^{-}} f(x)\) and \(\lim _{x \rightarrow 0^{+}} f(x)\)

Step-by-Step Solution

Answer

Question: Sketch a graph of the step function \(f(x) = \frac{|x|}{x}\) for \(x\neq0\) on the interval [-2,2], and determine if the limit of \(f(x)\) exists as \(x\) approaches 0. Solution: The step function's graph consists of two horizontal lines: \(y = 1\) for \(0 < x \leq 2\), and \(y = -1\) for \(-2 \leq x < 0\), with an open dot or hole at the points \((0, 1)\) and \((0, -1)\). As \(x\) approaches 0, the left-hand limit is \(-1\) and the right-hand limit is \(1\). Since they are not equal, the limit of the function does not exist.

Step 1: Understand the step function and differentiate cases

The step function is given by \(f(x) = \frac{|x|}{x}\) for \(x\neq0\). We will analyze the function for \(x > 0\) and \(x < 0\) separately. When \(x > 0\): \(f(x) = \frac{x}{x} = 1\) When \(x < 0\): \(f(x) = \frac{-x}{x} = -1\)

Step 2: Sketch a graph of the function on the interval [-2,2]

Based on the cases we analyzed before, we can sketch the function on the given interval: 1. For \(x > 0\): \(f(x) = 1\). Draw a horizontal line at \(y = 1\) for \(0 < x \leq 2\). 2. For \(x < 0\): \(f(x) = -1\). Draw a horizontal line at \(y = -1\) for \(-2 \leq x < 0\). 3. The function is not defined at \(x = 0\). Mark a hole or an open dot at the points \((0, 1)\) and \((0, -1)\).

Step 3: Examine the left-hand limit and right-hand limit of the function as \(x\) approaches 0

To determine if the limit of the function exists, we will analyze its left-hand limit and right-hand limit as \(x\) approaches 0. 1. Left-hand limit: We want to find \(\lim_{x\to 0^-}f(x)\). Since \(f(x) = -1\) for all \(x < 0\), the left-hand limit is equal to \(-1\). 2. Right-hand limit: We want to find \(\lim_{x\to 0^+}f(x)\). Since \(f(x) = 1\) for all \(x > 0\), the right-hand limit is equal to \(1\).

Step 4: Determine if the limit exists and explain the reasoning

Now we have the left-hand limit \(\lim_{x\to 0^-}f(x) = -1\) and the right-hand limit \(\lim_{x\to 0^+}f(x) = 1\). The limit \(\lim_{x\to 0}f(x)\) exists if and only if the left-hand limit is equal to the right-hand limit. In this case, the left-hand limit is not equal to the right-hand limit, so the limit \(\lim_{x\to 0}f(x)\) does not exist. The reason for this is that the function behaves differently on the left side of \(x = 0\) compared to the right side (\(f(x) = -1\) for \(x<0\) and \(f(x) = 1\) for \(x>0\)) and there is a hole or discontinuity at \(x = 0\).

Key Concepts

Limits of a Function

Understanding the concept of limits is fundamental in calculus, as it helps to analyze how a function behaves as the input approaches a certain value. In the context of the step function given by f(x) = frac{|x|}{x}, for x ≠ 0, we explore the limit as x approaches 0. A limit can exist if the function approaches a particular value from both the left and the right sides as x gets closer to the point of interest.

In simple terms, if you can predict the output of the function as you get infinitely close to a point from any direction, that’s the limit at that point. However, the limit does not exist if the function does not approach the same value from the left and the right. This is precisely the case with the given step function at x = 0, where the outputs for x approaching zero from either direction are different: -1 from the left and 1 from the right. This discrepancy in values clearly indicates that the limit at x = 0 for this function does not exist.

Graphing Functions

Graphing functions allows us to visualize the behavior of mathematical equations on a coordinate system. When dealing with a step function, such as f(x) = frac{|x|}{x}, for x ≠ 0, we separate the graph into different sections based on the function's behavior.

The function takes on only two values: -1 when x is negative, and 1 when x is positive. This forms two horizontal lines on the graph. It's important to note that at the point where x = 0, the function is undefined, which is usually represented by an open circle on the graph, signaling a discontinuity. Effective graphing requires careful consideration of these different behaviors to ensure an accurate representation of the function.

Left-hand Limit

The left-hand limit of a function at a certain point looks at how the function behaves as it approaches that point from the left side of the x-axis. Mathematically, we express this as lim_{x→c^-} f(x), where c is the point we are considering, and the minus superscript indicates the left-hand approach.

For the step function f(x) = frac{|x|}{x}, as x approaches 0 from the left (i.e., x becomes a smaller and smaller negative number), the value of f(x) remains at -1. This consistency gives us a left-hand limit of -1. Understanding left-hand limits is important, particularly when investigating if the overall limit exists at a discontinuity point.

Right-hand Limit

Complementing the left-hand limit is the right-hand limit. It measures the behavior of a function as the input approaches a specific point from the right side of the x-axis. In notation, we write this as lim_{x→c^+} f(x), with the plus symbol designating a right-side approach towards the point c.

In our exercise with the step function f(x) = frac{|x|}{x}, for x ≠ 0, the right-hand limit as x approaches zero is essential to determine if there is discontinuity. The function value stabilizes at 1 as x gets increasingly closer to zero from the right, giving us a right-hand limit of 1. This limit is significant for defining function behavior at points where it is not directly defined.