Problem 40 - Contemporary Precalculus

Question

A function \(f\) that is not defined at \(x=c\) is said to have \(a\) removable discontinuity at \(x=c\) if there is a function \(g\) such that \(g(c)\) is defined, \(g\) is continuous at \(x=c,\) and \(g(x)=f(x)\) for \(x \neq c .\) In Exercises \(39-43,\) show that the function \(f\) has a removable discontinuity by finding an appropriate function g. $$f(x)=\frac{x^{2}}{|x|}$$

Step-by-Step Solution

Answer

If so, show the appropriate function g(x) and provide your explanation. Answer: Yes, the function \(f(x)=\frac{x^{2}}{|x|}\) has a removable discontinuity. We find an appropriate function g(x) as follows: $$ g(x) = \begin{cases} x, & \text{ if } x > 0 \\ -x , & \text{ if } x < 0 \\ 0 , & \text{ if } x = 0 \\ \end{cases} $$ The function g(x) satisfies the condition g(x) = f(x) for \(x \neq 0\). Furthermore, g(0) is defined and equal to 0, which is the limit of f(x) as x approaches 0. Since g(x) is a continuous function, we have shown that f(x) has a removable discontinuity.

Step 1: Consider the function f(x)

We are given the function \(f(x) = \frac{x^{2}}{|x|}\). Notice that f(x) is undefined when x=0 since |x|=0 and we are dividing by the absolute value of x.

Step 2: Rewrite the function using a piecewise definition and find its limit as x approaches 0

We will first rewrite the function as a piecewise function, breaking it down into cases based on the positive or negative value of x: $$ f(x) = \begin{cases} \frac{x^2}{x} = x, & \text{ if } x > 0 \\ \frac{x^2}{-x} = -x , & \text{ if } x < 0 \\ \end{cases} $$ Now we will find the limit as x approaches 0: $$ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x = 0 \\ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (-x) = 0 $$ Since both the right-hand and left-hand limits are equal, we can say that the two-sided limit exists and is equal to 0: $$ \lim_{x \to 0} f(x) = 0 $$

Step 3: Define a new function g(x)

Now we can define a new function, g(x), that satisfies the condition \(g(x)=f(x)\) for \(x \neq 0\) and has g(0) = 0, since the limit of f(x) as x approaches 0 is 0. We can define g(x) as follows: $$ g(x) = \begin{cases} x, & \text{ if } x > 0 \\ -x , & \text{ if } x < 0 \\ 0 , & \text{ if } x = 0 \\ \end{cases} $$

Step 4: Show that g(x) satisfies the required conditions

We have defined g(x) such that g(x) = f(x) for \(x \neq 0\). Moreover, g(0) is defined and equal to 0, the limit of f(x) as x approaches 0. As g(x) is a continuous function (it is continuous at x=0 and for all other values of x), we have shown that f(x) has a removable discontinuity by finding an appropriate function g(x).

Key Concepts

Limit of a Function

In mathematics, the limit of a function describes the value that a function approaches as the input approaches a certain value. Limits are fundamental in calculus and analysis. Here, we explore the behavior of the function as the input approaches a specific point, which can help us understand continuous and discontinuous behavior at that point.
To find the limit of a function, we use notation like \( \lim_{x \to a} f(x) \). This means we are interested in what happens to \( f(x) \) as \( x \) gets very close to \( a \). If both the left-hand and right-hand limits are equal as \( x \to a \), the two-sided limit exists.

• Right-hand Limit: \( \lim_{x \to a^+} f(x) \)
• Left-hand Limit: \( \lim_{x \to a^-} f(x) \)

In the exercise, we found that both right-hand and left-hand limits of \( f(x) = \frac{x^{2}}{|x|} \) as \( x \to 0 \) are 0, so the two-sided limit is 0.

Piecewise Functions

Piecewise functions are defined by different expressions based on the input value. They're like conditional statements—different formulas for different parts of the domain. They are handy for describing situations where a rule changes depending on the value.
In our example, \( f(x) = \frac{x^{2}}{|x|} \) can be rewritten as a piecewise function:
\[ f(x) = \begin{cases}x, & \text{ if } x > 0 \-x, & \text{ if } x < 0 \\end{cases}\]
This decomposition helps understand different behavior across the domain and plays a crucial role when studying limits and continuity.

Continuity

A function is continuous if you can draw its graph without lifting the pen. More formally, a function \( f \) is continuous at a point \( c \) if:

• The limit \( \lim_{x \to c} f(x) \) exists
• \( f(c) \) is defined
• \( \lim_{x \to c} f(x) = f(c) \)

In our exercise, the function \( f(x) \) is not defined at \( x = 0 \), leading to a discontinuity. By creating a function \( g(x) \) that matches \( f(x) \) everywhere except at \( x = 0 \) and ensuring \( g(0) = 0 \), we've "removed" the discontinuity, as now \( g \) meets the continuity criteria at \( x = 0 \).

Function Definition

Function definition is crucial in mathematics, as it sets the rule for how input values are transformed into output values. This can appear in explicit algebraic formulas or piecewise definitions, depending on its consistency across a domain.
In our exercise, the function \( f(x) = \frac{x^{2}}{|x|} \) initially appears undefined at \( x = 0 \), as you cannot divide by zero. By redefining it piecewise, we clarify its behavior and prepare for examining limits and continuity.
A proper function definition helps anticipate places where problems like discontinuities could occur, aiding in better overall understanding of the function's behavior.

Precalculus Concepts

Precalculus serves as the foundation for calculus, including understanding functions, limits, and continuity. It involves concepts like transformations, asymptotes, and function's behavior, which are integral when studying calculus.
A deep understanding of these precalculus concepts is essential. Limits introduce how functions behave as inputs are near critical values, while continuity and discontinuous points show us where certain assumptions break.
Understanding piecewise functions further enriches this knowledge, allowing for flexible analysis of complex functions that cannot be described by a single formula. As seen in our example, seamlessly blending these ideas provides a robust framework needed for calculus.