Problem 100 - Calculus of a Single Variable

Question

Sketching a Graph Sketch the graph of any function \(f\) such that $$ \lim _{x \rightarrow 3^{+}} f(x)=1 \quad \text { and } \quad \lim _{x \rightarrow 3^{-}} f(x)=0 $$

Step-by-Step Solution

Answer

The function \(f\) should approach the value 0 as \(x\) becomes closer to 3 from the left and 1 as \(x\) becomes closer to 3 from the right. There should be a clear discontinuity at \(x=3\), with one 'portion' of the graph approaching (3,0) and the other approaching (3,1).

Step 1: Understand one-sided limits

The limit of a function as \(x\) approaches a certain value from either side can be different. This is denoted as \(x \rightarrow a^+\) and \(x \rightarrow a^-\). Here, at \(x=3^+\) the limit is 1 and at \(x=3^-\) the limit is 0. This suggests a point of discontinuity at \(x=3\).

Step 2: Scribble the initial sketch

Start by marking the point (3,0) as this is the limit of the function as \(x\) approaches 3 from the left. Similarly, mark the point (3,1) as this is the limit of the function as \(x\) approaches 3 from the right. Then, create two distinct lines or curves that approach these points as \(x\) approaches 3 from the left and right respectively.

Step 3: Finalize the Sketch

Examine the sketch and make sure it is consistent with the given conditions in the problem. The function should approach 0 as \(x\) approaches 3 from the left (depicting \(\lim_{x \rightarrow 3^-} f(x)=0\)), and it should approach 1 as \(x\) approaches 3 from the right (depicting \(\lim_{x \rightarrow 3^+} f(x)=1\)). The function should clearly indicate a point of discontinuity at \(x=3\).

Key Concepts

Limit of a Function

In calculus, the limit of a function is a fundamental concept that describes the behavior of the function as the input approaches a certain value. For instance, when we say \( \lim_{x \rightarrow c} f(x) = L \) what we're really talking about is what value the function \( f(x) \) is getting closer to when \( x \) gets closer to \( c \). Now, this doesn't always mean that \( f(x) \) will actually reach \( L \) when \( x \) is exactly \( c \), but rather that as \( x \) is nearly \( c \), \( f(x) \) is nearly \( L \) or asymptotically approaches \( L \).

This concept becomes particularly interesting when we consider one-sided limits. A one-sided limit tells us the value that a function approaches as the input approaches a certain value from only one side—either from the left \( (x \rightarrow c^-) \) or from the right \( (x \rightarrow c^+) \). When these one-sided limits are different, as in our exercise, it informs us that there is a distinctive behavior in the function at this specific point.

Point of Discontinuity

When one-sided limits at a point are not equal, this indicates a point of discontinuity. A point of discontinuity occurs where a function is not continuous, which means there is a sudden break or jump in the graph of the function. In the exercise, the function \( f \) has a discontinuity at \( x=3 \) because the left-hand limit \( \lim_{x \rightarrow 3^-} f(x) \) is 0 and the right-hand limit \( \lim_{x \rightarrow 3^+} f(x) \) is 1. These differing values implicate that there is no single value that \( f(x) \) approaches as \( x \) gets close to 3. This is a hallmark of a discontinuity, and graphically, it will look like a jump or gap at \( x=3 \) on the graph of \( f \).

Graphing Functions

Graphing functions is an essential skill in calculus for visualizing how a function behaves. An accurate graph can reveal much about a function's properties such as continuity, increasing and decreasing intervals, relative extrema, and points of discontinuity. Graphing the function from our exercise involves plotting points and understanding the behavior indicated by the limits. Start with sketching the left-hand behavior as \( x \) approaches 3 to show the function approaching the value of 0, then sketch the right-hand behavior as \( x \) approaches 3 to show the function approaching the value of 1. The graph should include a clear indication of the point of discontinuity at \( x=3 \) where these two behaviors do not align, signifying that the function does not have a single value at that point.

Calculus

Calculus is a branch of mathematics that studies change and motion. Within calculus, topics like limits, continuity, derivatives, and integrals play pivotal roles in understanding and analyzing the behavior of functions. Limits help us grasp the idea of approaching values, and they provide the foundational building blocks upon which the rest of calculus is built. Understanding the concepts of limits and discontinuities enables students to comprehend more advanced topics in calculus. Additionally, learning to visually interpret these concepts through graphing illuminates the comprehensive picture of a function's behavior and how it can be applied to real-world problems where change is a constant factor.