Problem 27 - Precalculus: Mathematical for Calculus

Question

Find the limit, if it exists. If the limit does not exist, èxplain why. $$\lim _{x \rightarrow-4}|x+4|$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Absolute Function

The function given is \( f(x) = |x + 4| \). This means for any value of \(x\), the function outputs the non-negative value of \(x + 4\).

Step 2: Substitute the Limit Point into the Function

The limit is to find how the function behaves as \(x\) approaches -4. Substitute \(x = -4\) into the function:\( f(x) = |-4 + 4| = |0| \).

Step 3: Simplify the Expression

The absolute value expression simplifies to \(|0| = 0\).

Step 4: Conclude on the Limit

As \(x\) approaches -4 from either direction, the function value approaches 0. Thus, the limit as \(x\) approaches -4 of the function is 0.

Key Concepts

Absolute Function

Absolute functions are essential mathematical tools that allow us to understand how a number behaves regardless of its sign. An absolute function is usually written as \(|x|\). What this notation represents is the absolute value of \(x\), which is:

• Leave \(x\) unchanged when \(x \geq 0\)
• Convert \(x\) to \(-x\) when \(x < 0\)

To grasp this better, think of the absolute function as measuring the "distance" a number is from zero on the number line. This measure is always positive, or zero, since distance cannot be negative. In the problem, the absolute function \(|x + 4|\) means we look at the positive value of \(x + 4\), even if \(x + 4\) is negative. The understanding that the absolute function outputs non-negative values helps us determine its limits as \(x\) approaches certain points.

Limit Point

The concept of a limit point is central to finding the limit of a function. A limit point is the value that \(x\) approaches within the function. In mathematical terms, we denote the limit point as \(x\to a\), where \(a\) is the point of interest.
When evaluating the limit \(\lim_{x \to -4}|x+4| \), our limit point here is \(-4\). This means we want to determine what the function \(|x + 4|\) becomes when \(x\) gets closer and closer to -4. By calculating and simplifying \(|x + 4|\) at \(x = -4\), we see that \(|-4 + 4| = |0| = 0\). Thus, the function at this limit point is zero.
In essence, a limit point helps give us an insight into the continuity or discontinuity of functions and their behavior around particular points.

Behavior of Functions

Understanding the behavior of functions is key to mastering limits. It involves analyzing how functions act as inputs approach specific values. In this problem, as \(x\) approaches \(-4\), we observe the function behavior through its output.
Since \(f(x) = |x+4|\) is a continuous absolute function, the behavior at \(x \to -4\) is smoothly predictable, leading to the value of 0.

• If the function's behavior is consistent from both sides of the limit point, we say the limit exists, as in our problem.
• If the function acts unpredictably or changes abruptly at the limit point, the limit may not exist.

Investigating the behavior of functions helps us in many situations. It tells us if limits are smooth or if there are potential complications like jumps or asymptotic behavior—features which might indicate a lack of limit at a specific point.