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A piecewise function is a type of mathematical function that is defined by different expressions depending on the value of the input variable. In simpler terms, the function can "switch" the formula used based on the condition of the input. This feature allows a function to behave differently on different parts of its domain.

In the given exercise, the function is defined as:

• For \(x > 0\), the function \(f(x) = \frac{|x|}{x} = 1\).
• For \(x < 0\), the function \(f(x) = \frac{|x|}{x} = -1\).

This demonstrates the piecewise nature of \(f(x)\), where each piece is a simple, constant function on either side of the origin.

At \(x = 0\), the expression \(\frac{|x|}{x}\) becomes undefined because division by zero is not possible. Thus, \(f(x)\) has different values on either side of the zero, reflecting its discontinuity there.

Limits help explain how the value of a function behaves as the input approaches a certain point. They are crucial in evaluating the behavior of functions at points where they might not be directly defined.

In the case of the function \(f(x) = \frac{|x|}{x}\), we want to understand its behavior near \(x = 0\). To do this, we calculate the right-hand limit and the left-hand limit separately:

• The right-hand limit is where \(x\) approaches zero from the positive side \( (x \to 0^+)\), resulting in \(\lim_{{x \to 0^+}} f(x) = 1\).
• The left-hand limit is where \(x\) approaches zero from the negative side \( (x \to 0^-)\), resulting in \(\lim_{{x \to 0^-}} f(x) = -1\).

Since the right-hand limit and left-hand limit are not equal, the limit of \(f(x)\) as \(x\) approaches 0 does not exist.

This discrepancy between the right and left limits is key to understanding why \(f(x)\) is not continuous at \(x = 0\).

The absolute value of a number is its distance from zero on the number line without considering direction. It’s denoted by two vertical bars surrounding the number, such as \(|x|\).

• For positive numbers, \(|x| = x\).
• For negative numbers, \(|x| = -x\) (which gives a positive result).

Absolute value is foundational in defining the piecewise function in our exercise.

Consider the function \(f(x) = \frac{|x|}{x}\). The behavior changes based on whether \(x\) is positive or negative.

• For \(x > 0\), \(|x| = x\), thus resulting in \(\frac{x}{x} = 1\).
• For \(x < 0\), \(|x| = -x\), thus resulting in \(\frac{-x}{x} = -1\).

This effect dramatically affects the function's continuity and shows the utility of absolute value in constructing piecewise functions that behave differently across their domains.