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The absolute value of a number essentially measures its distance from zero on the real number line. It is always non-negative, meaning it can be zero or positive but never negative. For a function \( f(x) \), the absolute value, denoted \( |f(x)| \), reflects how far the value of \( f(x) \) is from zero, disregarding any direction on the number line.

To break it down further:

• If \( f(x) \) is positive or zero, then \( |f(x)| = f(x) \).
• If \( f(x) \) is negative, then \( |f(x)| = -f(x) \), which makes it positive.

The concept of absolute value is crucial when working with limits because it lets us focus on the magnitude of a function without worrying about its sign. In the context of proving limits, like in this exercise, it ensures that we only look at the size of the values as \( x \) approaches \( c \), enhancing our understanding of the behavior of \( f(x) \).

The Squeeze theorem is a powerful tool in calculus, useful when the limit of a function is difficult to directly determine. It comes into play especially when you can identify two simpler functions that 'squeeze' a more complex function from above and below as \( x \) approaches a certain point.

For the Squeeze theorem to apply, you'll need:

• A function \( g(x) \) such that \( g(x) \leq f(x) \leq h(x) \) for all \( x \) near \( c \) (except possibly at \( c \) itself).
• Both \( \lim_{x \to c} g(x) = L \) and \( \lim_{x \to c} h(x) = L \).

When these conditions hold, the limit \( \lim_{x \to c} f(x) \) must also be \( L \).

In our exercise, \( |f(x)| \) is squeezed by 0 (below) and \( f(x) \) (above), because \( f(x) \to 0 \) as \( x \to c \). This ensures that \( \lim_{x \to c} |f(x)| = 0 \) using the Squeeze theorem. The theorem simplifies such proofs significantly by focusing on bounding behaviors.

Limit laws provide a set of rules that govern the behavior of limits of functions. They establish a foundation on which more complex limit problems can be solved, enabling us to handle a wide array of functions dreamily as they approach a particular point.

Here are some basic limit laws:

• Sum Law: \( \lim_{x \to c} (f(x) + g(x)) = \lim_{x \to c} f(x) + \lim_{x \to c} g(x) \)
• Product Law: \( \lim_{x \to c} (f(x) \cdot g(x)) = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x) \)
• Constant Law: \( \lim_{x \to c} k \cdot f(x) = k \cdot \lim_{x \to c} f(x) \)

Applying these laws simplifies the evaluation of limits by breaking down complex expressions into more manageable pieces.

In this particular exercise, we leverage the continuity-related aspect of limit laws. Specifically, the rule that the limit of the absolute value \( |f(x)| \) as \( x \to c \) is the same as the limit of \( f(x) \) as \( x \to c \), provided that limit exists. This foundational law helps conclude that since \( \lim_{x \to c} f(x) = 0 \), consequently, \( \lim_{x \to c} |f(x)| = 0 \) too, ensuring a seamless understanding of limit transitions.