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The absolute value function, denoted by \( f(x) = |x| \), is a fundamental concept in mathematics that reflects the size or magnitude of a number regardless of its sign. Imagine asking, "What is the distance of this number from zero on the number line?" That is what the absolute value represents.
\(|x|\) means:

• if \(x > 0\), \(|x| = x\)
• if \(x < 0\), \(|x| = -x\)
• if \(x = 0\), \(|x| = 0\)

This function is always non-negative since we consider it as a distance, and distance cannot be negative. For instance, while the number -3 is three units away from zero, its absolute value is written as \(|-3| = 3\).
In graphical form, the absolute value function forms a "V" shape, with its vertex at the origin (0,0). This significant shape is symmetrical around the y-axis, making it essential when considering limits.

The left-hand limit, expressed as \( \lim _{x \rightarrow 0^{-}} f(x) \), examines what happens to a function as \(x\) approaches a particular value from the negative direction. In simpler terms, it is the anticipated behavior of a function's outputs as \(x\) creeps closer to zero, but strictly from the negative side.
For the absolute value function \(f(x) = |x| \), let's consider approaching zero from the left. When \(x\) is negative, \(f(x) = -x\), because we take the positive of the negative \(x\) values. As these \(x\) values get steadily closer to zero, \(f(x)\) likewise approaches 0. Thus, we determine that \( \lim _{x \rightarrow 0^{-}} |x| = 0 \).
This straightforward observation ensures that the absolute value smoothly transitions through zero, supporting our understanding of the function's continuity from the left.

In contrast, the right-hand limit, \( \lim _{x \rightarrow 0^{+}} f(x) \), studies the function's behavior as \(x\) moves closer to a specific point from the positive side. This specific analysis presents clarity on how functions react from this direction.
For \(f(x) = |x| \), focusing on values nearing zero from the right, \(x\) remains positive. Thus, \(f(x) = x\). When \(x\) approaches zero from this side, \(f(x)\) also steadily moves towards zero. So, we conclude that \( \lim _{x \rightarrow 0^{+}} |x| = 0 \).
Both the left and right-hand limits meeting at zero allows us to say that the function \(f(x)=|x|\) is consistent and predictable when nearing zero, offering a clear picture of its behavior on the positive side.

The two-sided limit combines the insights from both one-sided limits: the left and right. For it to exist, \( \lim _{x \rightarrow a^{-}} f(x) = \lim _{x \rightarrow a^{+}} f(x) \) must hold true, producing one final value as \( \lim _{x \rightarrow a} f(x) \).
For the absolute value function at zero, we observe:

• The left-hand limit (originating from negative values) is 0.
• The right-hand limit (coming from positive values) is also 0.

Since both match, the two-sided limit is \( \lim _{x \rightarrow 0} |x| = 0 \). This verifies that the absolute value function is continuous at \(x = 0\).
This continuous outcome signifies no abrupt changes, highlighting why the absolute value function appears as a smooth "V" with a seamless transition across its vertex at the origin.