Problem 18 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(11-18,\) use the function \(f\) defined and graphed below to answer the questions. $$f(x)=\left\\{\begin{array}{ll}{x^{2}-1,} & {-1 \leq x<0} \\\ {2 x,} & {0 < x < 1} \\ {1,} & {x=1} \\ {-2 x+4,} & {1 < x < 2} \\ {0,} & {2 < x < 3}\end{array}\right.$$ Writing to Learn Is it possible to extend \(f\) to be continuous at \(x=3 ?\) If so, what value should the extended function have there? If not, why not?

Step-by-Step Solution

Answer

No, it is not possible to extend the function \(f\) to be continuous at \(x = 3\) due to the absence of a right-hand limit at this point.

Step 1: Determine the left-hand limit

The left-hand limit of a function as \(x\) approaches \(a\) is the value the function approaches as \(x\) comes to \(a\) from the left side. It is denoted as \(\lim_{{x \to a^-}} f(x)\). The relevant piece of the function as \(x\) approaches 3 from the left is \(f(x) = 0\) for \(2 < x < 3\). So the left-hand limit as \(x \to 3\) is 0.

Step 2: Determine the right-hand limit

The right-hand limit as \(x\) approaches \(a\) is the value the function approaches as \(x\) comes to \(a\) from the right side, denoted as \(\lim_{{x \to a^+}} f(x)\). Since we have no definition of the function for \(x > 3\), there is no right-hand limit.

Step 3: Consider the continuity at x = 3

A function is continuous at \(x = a\) if three conditions are met: \(\lim_{{x \to a^-}} f(x) = f(a)\), \(\lim_{{x \to a^+}} f(x) = f(a)\) and \(f(a)\) is defined. We have seen that the left-hand limit at \(x = 3\) is 0, there is no right-hand limit, and \(f(3)\) is not defined. However, given that we are asked to extend the function to \(x = 3\), we can define \(f(3) = 0\). This would make the function match the left-hand limit at \(x = 3\), but there is still no right-hand limit. Thus, even with this extension, the function is not generally considered to be continuous at \(x = 3\), as it fails to have a well-defined right-hand limit.

Key Concepts

Left-hand Limit

When we talk about the left-hand limit of a function, we refer to the value that the function approaches as the variable approaches a certain number from the left side. Mathematically, if we are examining the limit at a point 'a', the left-hand limit is denoted as \( \lim_{{x \to a^-}} f(x) \). To find this limit, we only consider the part of the function that is defined to the left of 'a'.

For instance, in the provided problem where the function is defined over various intervals, the left-hand limit at \( x = 3 \) is approached by looking at the interval \( 2 < x < 3 \) where \( f(x) = 0 \). Consequently, as \( x \) gets closer and closer to 3 from the left, the value of the function \( f(x) \) remains constant at zero. Thus, the left-hand limit as \( x \to 3 \) is indeed 0.

Right-hand Limit

Conversely, the right-hand limit is concerned with the behavior of a function as the variable approaches a number from the right. Denoted by \( \lim_{{x \to a^+}} f(x) \), it requires analyzing the portion of the function that lies to the right of 'a'.

In the exercise, there is no function defined for values greater than 3, which means the right-hand limit at \( x = 3 \) does not exist since there is no behavior to analyze. This absence is significant because for a function to be continuous at a point, we require both left-hand and right-hand limits to exist and be equal.

Piecewise Functions

The function in the exercise is an example of a piecewise function. These functions are special because they have different expressions or 'pieces' for different intervals of the independent variable. The different 'pieces' mean that the function's formula changes based on the input value of \( x \).

Each 'piece' is defined for a specific interval, as seen with the function \( f(x) \) that has separate definitions for the intervals \( -1 \leq x < 0 \) and \( 0 < x < 1 \) and so on. Understanding each piece is crucial for exploring limits and evaluating the continuity of a piecewise function at the points where the formula changes.

Limits of a Function

The concept of limits underpins much of calculus and specifically informs the understanding of continuity. The limit of a function at a given point describes the value that the function approaches as the input variable approaches a specific value. It's a way of describing behavior near a point, not necessarily at that point.

Mathematically, this is shown as \( \lim_{{x \to a}} f(x) \), which reads as 'the limit of \( f(x) \) as \( x \) approaches \( a \).' In context, if we can find a value that \( f(x) \) gets arbitrarily close to as \( x \) approaches \( a \) from either side, then the limit exists at that point. If \( f(x) \) approaches different values from the left and the right, or if it doesn't approach any value at all, then the limit does not exist.

Defining Continuity

Finally, let's dive into defining continuity. A function is said to be continuous at a point \( x = a \) if three conditions are met: the function is defined at \( a \) (\( f(a) \) is defined), the limit as \( x \) approaches \( a \) exists (\( \lim_{{x \to a}} f(x) \) exists), and the limit equals the function's value at that point (\( \lim_{{x \to a}} f(x) = f(a) \) ).

In our exercise, defining \( f(3) = 0 \) partially aligns with the requirements for continuity since it matches the left-hand limit, but because the right-hand limit does not exist, we cannot say that the function is continuous at \( x = 3 \). It illustrates a crucial aspect of continuity: the function must be predictable and follow a clear trend on either side of the point in question.