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One-sided limits are used to examine the behavior of functions as they approach a specific point from one direction only. They help us understand the function's behavior near that point without crossing it. In calculus, these are crucial because they can determine how a function behaves at points where it might not be continuous.

For example, when dealing with the function \(\lim _{x \rightarrow 1^{+}} \, f(x)\), it considers the behavior of \(f(x)\) as \(x\) approaches 1 from the right (positive side). This is distinct from:

• \(\lim _{x \rightarrow 1^{-}} f(x)\), examining from the left (negative side).

These distinctions are vital when solving limits close to points of discontinuity or abrupt changes, as we see in piecewise functions. Understanding one-sided limits helps identify situations where these do not match, implying that a two-sided limit doesn't exist.

If the two-sided limit exists at a point, the one-sided limits from both directions must equal each other at that point.

Piecewise functions are functions defined by multiple sub-functions, each applied to a specific interval of the domain. These are very useful for modeling scenarios where behavior changes at certain intervals. Consider them like a path with different segments, each with a unique direction.

With piecewise functions, it's crucial to consider:

• The specific formula used within each segment
• The endpoints where segments meet
• Continuity at these endpoints

In this exercise, the absolute value function introduces a piecewise element. The expression involves \(|x-1|\), which behaves differently based on whether \(x\) approaches 1 from above or below. This means two separate rules are applied, leading to different behavior analyzed through one-sided limits.

Fully understanding piecewise functions helps in determining the overall behavior of a compound function, ensuring continuity and evaluating limits correctly.

The absolute value function is a critical concept in calculus and is often expressed as \(|x| = x\) if \(x \geq 0\) and \(-x\) if \(x < 0\). It represents the distance of a number from zero on the number line, always yielding non-negative results.

In limit problems, the absolute value becomes essential when handling expressions like \(|x-1|\). Here, the behavior of the absolute value changes:

• If \(x > 1\), then \(|x-1| = x-1\)
• If \(x < 1\), then \(|x-1| = -(x-1)\)

This piecewise analysis helps distinguish different values or expressions for cases approaching from either side. When examining limits, these variations can cause inequalities between one-sided limits—a vital aspect to note.

Mastering absolute value analysis aids in correctly setting up and simplifying expressions, a fundamental skill to solve calculus problems efficiently.