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The limit of a function deals with what happens to a function value as its input approaches a particular point. In formal terms, the limit of a function \(f(x)\) as \(x\) approaches \(c\) is \(L\) if for every number \(\epsilon > 0\), there exists a number \(\delta > 0\) such that whenever \(0 < |x - c| < \delta\), it follows that \(|f(x) - L| < \epsilon\). This is a precise way of saying, "As \(x\) gets closer to \(c\), \(f(x)\) gets closer to \(L\)."
Understanding this concept is crucial because it provides a rigorous foundation for many other mathematical topics. The idea of a limit underlies the definition of derivatives and integrals, which are the building blocks of calculus.
To prove that a limit is indeed \(L\), we must show the existence of such \(\epsilon\) and \(\delta\) for all possible values of \(\epsilon\). This process confirms that no matter how close you want \(f(x)\) to get to \(L\), you can always find a small enough \(\delta\) to make it happen.

The absolute value of a number is essentially its distance from zero on the number line. Important properties of absolute value include:

• \(|a| = a\) if \(a \geq 0\)
• \(|a| = -a\) if \(a < 0\)
• \(|a| \geq 0\) for all real numbers \(a\)
• \(|a| \geq a\) and \(|a| \geq -a\)

These properties are crucial for understanding inequalities involving absolute values.
In our example, to prove \(\lim_{x \rightarrow c} |f(x)| = 0\), we rely on the property that \(|f(x)| = |f(x) - 0|\). This ensures that the distance of \(f(x)\) from zero is not greater than the distance of \(f(x)\) from zero itself, making it necessary for \(|f(x)|\) to also approach zero as \(f(x)\) does. Hence, knowing these properties helps establish the connection between regular functions and their absolute counterparts.

The epsilon-delta definition of a limit is a formal way to define the concept of limits in calculus, providing a robust framework to work with. This definition is how mathematicians ensure their claims about limits are accurate and precise.

In this framework, for a function \(f(x)\) to have a limit \(L\) as \(x\) approaches \(c\), it must satisfy that for every number \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x - c| < \delta\), then \(|f(x) - L| < \epsilon\). This approach essentially breaks down to finding \(\delta\) that works for any "target closeness" \(\epsilon\).

In the proof that \(\lim_{x \rightarrow c} f(x) = 0\) implies \(\lim_{x \rightarrow c} |f(x)| = 0\), the epsilon-delta definition ensures the transformation from \(f(x)\) to \(|f(x)|\) still holds the limit at 0. This is a pivotal step in the proof, using the understanding that whatever holds for \(f(x)\), in terms of closeness to zero, will inherently hold for \(|f(x)|\) using absolute value properties.