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Absolute value functions take any input, often a variable inside an expression like \(|x+5|\), and treat it as a non-negative number. This means that whatever value is inside, the absolute value function converts it to a positive number or leaves a positive number unchanged. This characteristic plays a crucial role in breaking down the function into two separate cases based on whether the expression behaves positively or negatively.
When we have functions like \(\frac{|x+5|}{x+5}\), the absolute value affects the numerator only and turns the expression \(x+5\) into a positive value no matter what.
However, to simplify it further, we look into two scenarios:

• When \(x+5\) is positive, \(|x+5| = x+5\).
• When \(x+5\) is negative, \(|x+5| = -(x+5)\).

This split allows us to investigate the behavior of limits concerning absolute value.

The left-hand limit focuses on what happens to the function as the variable approaches a certain value from the left side. In our exercise with the function \(\frac{|x+5|}{x+5}\), we observe what happens as \(x\) comes closer to \(-5\) from values less than \(-5\).
For \(x < -5\), the expression \(x+5\) is negative, so \(|x+5| = -(x+5)\). Plugging this into our function, we find that it simplifies to \(-1\). As \(x\) approaches \(-5\) from the left, the limit is determined as \(-1\).
This is our left-hand limit, indicated symbolically as \(\lim_{x \rightarrow -5^-}\frac{|x+5|}{x+5} = -1\). Evaluating the function from the left helps understand the function's behavior close to the point of interest.

The right-hand limit evaluates the function as \(x\) approaches a specific value from the right side. For our function \(\frac{|x+5|}{x+5}\), we look what happens when \(x\) nears \(-5\) from values greater than \(-5\).
When \(x > -5\), the expression \(x+5\) is positive. Thus, the absolute value \(|x+5|\) equals \(x+5\), leaving the function to simplify to \(1\). As \(x\) comes closer to \(-5\) from the right, the function approaches \(1\).
This result is known as the right-hand limit, symbolized by \(\lim_{x \rightarrow -5^+}\frac{|x+5|}{x+5} = 1\). Understanding the right-hand limit allows us to see how function values behave when inputs get infinitely close to the boundary from one direction.

Discontinuity in a function occurs when the function makes a 'jump' or change unexpectedly rather than following smoothly. This typically happens when the left-hand limit and right-hand limit are not equal as you approach a given point.
In our exercise example, this discontinuity takes place at \(x = -5\). As \(x\) approaches \(-5\), the left-hand limit is \(-1\), but the right-hand limit is \(1\).
These differing results tell us that there is a discontinuity at \(x = -5\). Since the limits from both sides are not equal, the overall limit at that point does not exist. Discontinuity is important because it affects how a function is understood especially in mathematical modeling and calculus.

Undefined points in a function are specific points where the function does not yield a valid output. This usually occurs when the function involves division by zero or other indeterminate forms.
In our scenario with the function \(\frac{|x+5|}{x+5}\), the function becomes undefined at \(x = -5\) because it involves dividing by zero. This results in the whole function being undefined at that point.
Mathematically, undefined points are key to understanding where and how a function behaves or fails to exist. Identifying these points helps in analyzing the continuity and validity of a function within a certain domain or while solving limit problems.