91Ó°ÊÓ

Piecewise functions are unique in that they are defined by multiple expressions over different intervals of the domain. In this exercise, the function \( f(x) \) is defined in two parts: for \( x \leq 0 \), the expression is \( x^2 \), and for \( x > 0 \), the expression is \( 1 - x^2 \). Piecewise functions often require separate analysis in different sections of their domains.

This structure allows a function to behave differently over various intervals, like being one type of polynomial in one range and another elsewhere. When dealing with limits of piecewise functions, it is crucial to choose the correct piece for evaluation based on the limit point and any conditions like \( x > 0 \) or \( x \leq 0 \).

Understanding which part of the function to use requires recognizing where your point of interest (in this case, \( x = 3 \)) falls in the domain intervals given by the function. Since 3 is greater than 0, we use the second piece, \( 1-x^2 \), to evaluate the limit.

The rigorous definition of a limit is fundamental in calculus and provides a precise way of describing the behavior of functions as the input approaches a particular value. For a function \( f(x) \), the limit \( \lim_{x \to c} f(x) = L \) if, for every positive number \( \epsilon \), there exists a positive number \( \delta \) such that whenever \( 0 < |x - c| < \delta \), it follows that \( |f(x) - L| < \epsilon \).

This ensures that as \( x \) gets arbitrarily close to \( c \), the values of \( f(x) \) approach \( L \) within any desired level of accuracy. The delta-epsilon definition is a mathematical way to capture the notion that we can make \( f(x) \) as close as we like to \( L \) by making \( x \) sufficiently close to \( c \).

When analyzing piecewise functions with limits, we apply this definition specifically to the relevant piece of the function at the point we're investigating. This exercise shows that by using \( f(x) = 1 - x^2 \) and seeing \( \lim_{x \to 3} f(x) = -8 \), it's possible to confirm the closeness condition by choosing appropriate \( \delta \) as required by the definition.

Evaluating limits is about finding where the function is heading as the input approaches a certain value. In piecewise functions, this involves determining which part of the function to use. When evaluating \( \lim_{x \rightarrow 3} f(x) \), since \( x > 0 \) when approaching 3, we apply the expression \( 1 - x^2 \).

To compute this limit directly, substitute \( x = 3 \) into the appropriate function part, giving \( 1 - 3^2 = 1 - 9 = -8 \). This provides the limit value straightforwardly, but confirmation through the rigorous definition using the epsilon-delta approach ensures accuracy by showing continuity over intervals near 3.

Successfully finding limits involves attentively examining conditions given by the function's definition, and applying the correct part to smoothly evaluate as \( x \) approaches the desired point, supported by theoretical backing like the delta-epsilon criteria to affirm the accuracy of the limit value. This double-check assures that no unexpected behavior or discontinuity at the approaching juncture falsely suggests the wrong limit.