Problem 33 - Precalculus: Mathematics for Calculus

Question

A Function with Specified Limits Sketch the graph of an example of a function \(f\) that satisfies all of the following conditions. $$\begin{aligned} &\lim _{x \rightarrow 0^{-}} f(x)=2 \quad \lim _{x \rightarrow 0^{+}} f(x)=0\\\ &\lim _{x \rightarrow 2} f(x)=1 \quad f(0)=2 \quad f(2)=3 \end{aligned}$$ How many such functions are there?

Step-by-Step Solution

Answer

There are infinitely many functions that satisfy these conditions.

Step 1: Understanding the Problem

We need to create a function that satisfies certain limit conditions and specific function values. The limits will dictate the behavior of the graph as it approaches specific points.

Step 2: Interpreting Limit at x=0

The limit \( \lim_{{x \rightarrow 0^{-}}} f(x)=2 \) means as \( x \) approaches 0 from the left, \( f(x) \) needs to approach 2. Meanwhile, \( \lim_{{x \rightarrow 0^{+}}} f(x)=0 \) indicates that as \( x \) approaches 0 from the right, \( f(x) \) needs to approach 0. This implies a jump discontinuity or a sharp turn at \( x=0 \).

Step 3: Interpreting Function Value at x=0

Since \( f(0) = 2 \), at exactly \( x=0 \), the function value is 2. This confirms the jump discontinuity hinted by the previous step.

Step 4: Interpreting Limit at x=2

The limit \( \lim_{{x \rightarrow 2}} f(x)=1 \) implies that as \( x \) approaches 2 from either direction, \( f(x) \) must approach 1. However, this does not mean the function necessarily equals 1 at \( x=2 \).

Step 5: Interpreting Function Value at x=2

We know that \( f(2)=3 \). This means that at \( x=2 \), the function differs from its approaching value, indicating a removable or point discontinuity at \( x=2 \).

Step 6: Sketching a Possible Function

Considering all the above conditions, sketch a function that has a sharp edge at \( x=0 \), with values approaching 2 from the left and 0 from the right, and facing a different value at \( x=2 \) than approaching but with the point value of 3.

Step 7: Assessing the Number of Possible Functions

There are infinitely many functions that can satisfy these conditions, as long as they meet the specified limits and function values exactly at \( x=0 \) and \( x=2 \).

Key Concepts

Jump Discontinuity

When discussing limits and graphing functions, one might encounter a jump discontinuity. This term describes a scenario where the graph of a function experiences an abrupt leap from one value to another as the input approaches a certain point from different directions. In simpler terms, imagine the graph 'jumping' to a different height without any gradual change.

Consider our example in this exercise:

• As \( x \) approaches 0 from the left, \( f(x) \) approaches 2 (\( \lim_{{x \rightarrow 0^{-}}} f(x)=2 \)).
• However, approaching from the right, \( f(x) \) approaches 0 (\( \lim_{{x \rightarrow 0^{+}}} f(x)=0 \)).

Thus, as you approach zero from both sides, the function's value shifts abruptly. Such behavior leads to a jump discontinuity because at exactly \( x = 0 \), we actually have \( f(0) = 2 \). The graph will show a distinct break or gap at this point.
This discontinuity is visually represented on a graph by a sharp change or a sudden 'jump' and is quite common when encountering limits.

Removable Discontinuity

A removable discontinuity is another type of imperfection that can occur on a graph. It resembles a 'hole' where the graph could have been seamlessly connected, yet isn't, due to a specific function value different from the approaching limits.

In the exercise:

• We see that as \( x \) approaches 2, \( f(x) \) closes in on 1, indicated by \( \lim_{{x \rightarrow 2}} f(x)=1 \).
• However, the function value at \( x=2 \) is defined as 3 (\( f(2) = 3 \)).

This deviation from the approaching value implies a removable discontinuity. With such a discontinuity, you could imagine 'removing' or altering the point at \( x=2 \) to align it with the approaching limits on the graph. This 'hole' in the continuity can often be 'fixed' by redefining a single point, although the original values define it distinctly in this case. Removable discontinuities are crucial in understanding how closely functions can approximate continuous behavior, even with minor discontinuity.

Sketching Graphs

Creating a visual representation of function behaviors through sketching graphs is a vital part of understanding limits and discontinuities. It allows you to see where discontinuities occur and how the function behaves around those points.

In this exercise, we have:

• A sharp change at \( x=0 \), where the function jumps from one side to another, typical of a jump discontinuity.
• A different value where the function meets at \( x=2 \) despite its limits, indicating a removable discontinuity.

When sketching:- Start by marking the specified points where limits and function values are given.- Indicate the approaching values using dashed lines or arrows, showing where the graph should approach from each direction.- Use circles to illustrate open points, where applicable.By visualizing these behaviors, you can better understand the significance of discontinuities in function limits. Sketching these graphs solidifies your comprehension and helps identify infinite possibilities of function forms that satisfy the given criteria.