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A piecewise function is a function defined by different expressions based on certain intervals of the independent variable, typically denoted as \(x\). These functions are useful to model situations where a process might change abruptly at certain points, like tax brackets or shipping costs based on weight.
For example, in our problem, the function \( f(x) \) is defined as:

• \( f(x) = x \) when \( x < 0 \)
• \( f(x) = \frac{1}{x} \) when \( x > 0 \)

Each formula is used to calculate \( f(x) \) in its respective interval. This indicates that our function could behave differently when approaching a particular point from different directions. Understanding how these separate parts behave near specific points is crucial to evaluating limits of piecewise functions.

The limit from the left of a function, denoted as \( \lim_{x \to a^-}f(x) \), refers to the value that the function \( f(x) \) approaches as \( x \) gets infinitely close to \( a \) from the left side.
In our scenario, we are analyzing \( f(x) \) as \( x \) approaches zero from the left, denoted as \( \lim_{x \to 0^-} f(x) \). Since \( f(x) = x \) whenever \( x < 0 \), its value becomes closer and closer to zero as \( x \) approaches zero. Hence, \( \lim_{x \to 0^-} f(x) = 0 \).
In simpler terms, we can say that approaching zero from numbers less than zero (like -0.1, -0.01) will make the output of the function approach zero itself.

The limit from the right, noted as \( \lim_{x \to a^+}f(x) \), is the value \( f(x) \) approaches as \( x \) gets infinitely close to \( a \) from the right side.
In this particular task, we consider \( f(x) \) as \( x \) approaches zero from the right: \( \lim_{x \to 0^+} f(x) \). For \( x > 0 \), \( f(x) = \frac{1}{x} \). As \( x \) approaches zero from the right, \( \frac{1}{x} \) grows without bound and heads towards \( \infty \).
This means that as \( x \) comes closer to zero from positive numbers (like 0.1, 0.01), the value of \( f(x) \) increases infinitely.

In calculus, a limit is defined at a point if the function approaches the same value from both the left and the right. For \( \lim_{x \to a} f(x) \) to exist, it's necessary that \( \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) \).
In the problem given, we determined that \( \lim_{x \to 0^-} f(x) = 0 \) and \( \lim_{x \to 0^+} f(x) = \infty \). Since these two results are not equal, the overall limit \( \lim_{x \to 0} f(x) \) cannot exist.
Consequently, when the left and right limits differ, it indicates that there is a discontinuity at that point, which implies the absence of a finite overall limit. This is a common occurrence in piecewise functions where different rules apply on either side of a point of interest.