Problem 13 - Calculus Early Transcendentals

Question

Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions. $$\lim _{x \rightarrow 1^{-}} f(x)=2, \quad \lim _{x \rightarrow 1^{+}} f(x)=-2, \quad f(1)=2$$

Step-by-Step Solution

Answer

The function approaches 2 from the left and -2 from the right, with f(1) = 2.

Step 1: Understanding the Limits

We have two different limits as we approach 1 from the left (\( x \to 1^- \)) and from the right (\( x \to 1^+ \)). For \( x \to 1^- \), the limit is 2, which means that the function approaches the value 2 from the left of 1. For \( x \to 1^+ \), the limit is -2, indicating the function approaches -2 from the right of 1.

Step 2: Function Value at x = 1

The function is defined such that \( f(1) = 2 \). This gives a point on the graph at (1, 2). Whatever behavior the function has when approaching from the left or the right, the point (1, 2) is firmly fixed on the graph.

Step 3: Sketching the Graph

To meet the conditions, sketch a graph that approaches 2 as \( x \to 1^- \), creating a curve or line approaching from the left towards the point (1, 2). From the right, draw a curve that approaches -2 as \( x \to 1^+ \), showing a discontinuity at x = 1. Place a solid dot at (1, 2) to represent \( f(1) = 2 \), despite the jump discontinuity where the graph approaches different y-values from the left and the right as x approaches 1.

Key Concepts

Discontinuity in Functions

A discontinuity in a function is where there is an 'interruption' or a 'jump' in the graph of the function. It means the function is not perfectly smooth or connected at that point. In the context of our example, the function has a discontinuity at the point where \( x = 1 \).

This is because as \( x \) approaches 1:

• From the left (denoted \( x \to 1^- \)), the function gets closer and closer to the value 2.
• From the right (\( x \to 1^+ \)), the function instead approaches the value -2.

Since these two directional limits point to different values, the function cannot smoothly connect at \( x = 1 \), hence it's discontinuous at that point. This is a classic example of a 'jump discontinuity', where the 'jump' happens because the left-side and right-side limits do not equal each other.

One-Sided Limits

One-sided limits give detailed information about a function's behavior as it approaches a particular point. They allow us to examine what happens from one specific direction, either from the left or the right.

In our exercise, we have two one-sided limits as we approach \( x = 1 \):

• The left-hand limit, \( \lim_{x \to 1^-} f(x) = 2 \), examines the function as \( x \) approaches 1 from the left side.
• The right-hand limit, \( \lim_{x \to 1^+} f(x) = -2 \), looks at \( x \) approaching 1 from the right side.

These calculations are essential for understanding the complete behavior of a function around critical points. Without computing both one-sided limits, we might miss features like the discontinuity present in this case.

Graph Sketching

Graph sketching is a valuable skill for visualizing and understanding the behavior of functions. By sketching the graph, we can quickly observe key details such as where a function is continuous or discontinuous, its limits, and specific values.

For this problem, you need to sketch a graph where:

• The graph approaches the value 2 as \( x \to 1^- \), representing the left-hand side limit.
• The graph approaches the value -2 as \( x \to 1^+ \), representing the right-hand side limit.
• There is a solid dot at the point (1, 2), showing that \( f(1) = 2 \), and marking the point despite the jump in the graph.

By accurately sketching such features, we visualize the complete structure of a function's behavior at \( x = 1 \), highlighting how it reacts differently from either side and the exact nature of its discontinuity.