91Ó°ÊÓ

Understanding factorization is a fundamental step in simplifying mathematical expressions, especially when dealing with complex functions. In the given exercise, we have the function \[g(x) = \frac{x^2 + x - 6}{|x-2|}\].
To simplify this expression, we need to factor the numerator, which is a quadratic expression.
To factor \(x^2 + x - 6\), we look for two numbers that multiply to \(-6\) and add to \(1\). These numbers are \(-2\) and \(3\). Thus, the factorization of the quadratic is \((x-2)(x+3)\).
With this factorization, the expression can be rewritten, helping further evaluation and simplification when analyzing limits. This process is essential as we approach the points of interest around \(x = 2\), where the denominator includes \(|x-2|\).

Piecewise functions consist of different expressions depending on the domain specified. In this exercise, \(g(x)\) is actually a piecewise function due to the presence of \(|x-2|\) in the denominator.
When considering \(x > 2\), we know \(|x-2| = x-2\). This simplification leads the function \(g(x)\) to behave as \(x+3\). However, for \(x < 2\), we have \(|x-2| = -(x-2)\), which makes the function \(-1 imes (x+3)\).
These different expressions work together to create a piecewise function:

• For \(x > 2\), function is \(x + 3\)
• For \(x < 2\), function is \(-(x + 3)\)

This piecewise nature is key to understanding the function's behavior around the point where \(x = 2\).

Discontinuity occurs when a function is not continuous at a given point. In the context of this exercise, we look at the behavior of \(g(x)\) at \(x = 2\) to determine if any discontinuity exists.
The limit analyses for both left-hand and right-hand approaching \(x = 2\) show different results:

• For \(\lim_{x \to 2^+} g(x) = 5\)
• For \(\lim_{x \to 2^-} g(x) = -5\)

Because these two values are not equal, there exists a discontinuity at \(x = 2\). This particular type of discontinuity is known as a jump discontinuity. It signifies that the function does not transition smoothly and instead 'jumps' from one value to another at that point.
Understanding this discontinuity is vital while analyzing or sketching the function's graph for intuitive learning.

Graph sketching involves creating a visual representation of a function over its domain. For \(g(x)\), knowing its piecewise behavior and discontinuity at \(x = 2\) plays a crucial role.
To sketch the graph:

• Start with the portion \(x + 3\) for \(x > 2\). This is a straight line with slope \(1\) intersecting the y-axis at \(3\).
• Next, graph the part \(-(x + 3)\) for \(x < 2\). This is also a line but slopes downwards, creating a mirror image of \(x + 3\).

Crucially, at \(x = 2\), there is a jump due to the discontinuity. The graph of \(g(x)\) decomposes into two parts that do not meet at this point, depicting a sharp transition. Observing and sketching around this point aids in grasping the transition and evaluating how each part of the piecewise function forms the overall picture.