91Ó°ÊÓ

The absolute value function, denoted as \(|x|\), is a fundamental concept in mathematics. It is used to measure the magnitude or size of a number, without considering its direction or sign. In simpler terms, the absolute value of a number is always non-negative. For any real number \(x\), the absolute value \(|x|\) can be defined as follows:

• If \(x\) is positive or zero, \(|x| = x\).
• If \(x\) is negative, \(|x| = -x\).

This definition ensures that we are always measuring the "distance" of a number from zero, regardless of its sign. \(|x|\) reflects how far away a number is from zero on the number line, treating positive and negative values as equal in magnitude. Understanding this concept is crucial when analyzing limits, as it affects how functions behave as they approach a particular value.

The left-hand limit of a function refers to the value that the function approaches as the independent variable approaches a specific point from the negative side, or left. When we consider the left-hand limit for the absolute value function \( |x| \) as \(x\) approaches 0, we specifically look at values where \(x < 0\).
For these negative values, the expression for \( |x| \) becomes \( -x \). As \( x \) gets closer to 0 from the left, \( -x \) also approaches 0.

Mathematically, this is expressed as:

• \( \lim_{x \to 0^-} |x| = \lim_{x \to 0^-} (-x) = 0 \).

This means that, from the left, the absolute value of \(x\) tends to zero, aligning with our intuitive understanding that as you approach zero from the negative side, the magnitude of those negative numbers shrinks to zero.

Similarly, the right-hand limit refers to the behavior of a function as the variable approaches a specific value from the right, or positive side. In our case, we evaluate the right-hand limit of the absolute value function \( |x| \) as \(x\) approaches 0 from the positive direction.
For positive values of \(x\), \( |x|\) simplifies directly to \(x\).

As \( x \) approaches 0 from the right side, the value of \( |x|\) or simply \( x \) gets closer to zero. This leads to the following mathematical representation:

• \( \lim_{x \to 0^+} |x| = \lim_{x \to 0^+} x = 0 \).

This indicates that from the positive direction, the distance of \(x\) from zero continues to diminish, reaching zero, which again confirms our understanding of how limits work.

A two-sided limit is considered when the function approaches a particular value from both the left and the right, and both one-sided limits are equal. In simpler terms, if the function reaches the same value from both directions at a point, then the two-sided limit exists.
In this exercise, we have found both the left-hand and right-hand limits as \(x\) approaches 0 for the absolute function \( |x| \), and both limits are equal to zero:

• Left-hand limit: \( \lim_{x \to 0^-} |x| = 0 \).
• Right-hand limit: \( \lim_{x \to 0^+} |x| = 0 \).

Since both sides result in the same value, we can conclude that the two-sided limit at 0 exists, and it equals zero.Therefore, the overall limit of the function as \(x\) approaches 0 is given by:

• \( \lim_{x \to 0} |x| = 0 \).

This reinforces the consistent nature of the absolute value function nearing zero from any direction on the number line.