Problem 28 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The limit exists and is -1.

Step 1: Understanding the Problem

We need to determine the limit of \( \frac{|x+4|}{x+4} \) as \( x \rightarrow -4^- \). This expression involves an absolute value, which can affect the limit depending on the direction from which \( x \) approaches \(-4\). Here, we approach from the left, i.e., \( x < -4 \).

Step 2: Simplifying the Expression

For \( x < -4 \), the expression \( x+4 \) is negative. Thus, the absolute value \( |x+4| = -(x+4) \) (because \( |x+4| = x+4 \) if it were positive, which is not the case here). The expression simplifies to \(-1\) because: \[ \frac{|x+4|}{x+4} = \frac{-(x+4)}{x+4} = -1 \].

Step 3: Evaluating the Limit

Since the expression simplifies to \(-1\) for all \( x < -4 \), the limit is constant. Therefore, as \( x \rightarrow -4^- \), the limit of the function is \(-1\).

Key Concepts

Absolute Value

When dealing with limits, the absolute value function can sometimes cause confusion, especially if the expression inside the absolute value approaches zero. Absolute values eliminate any negative sign, converting them into positives. It is denoted by vertical bars. For instance,

• \(|x| = x\) if \(x \geq 0\) because it’s already non-negative.
• However, \(|x| = -x\) if \(x < 0\) to make negative values positive.

In the given expression, \(|x+4|\) requires special attention because it depends on whether \(x+4\) is positive or negative.
When \(x\) is approaching from the left of \(-4\), \(x+4\) is negative, making \(|x+4| = -(x+4)\). This is crucial because the limit might differ depending on the direction from which \(x\) approaches the point of interest.

One-Sided Limits

One-sided limits are key when evaluating the behavior of a function as \(x\) approaches a certain value, but strictly from one direction—either from the left or the right.
This technique is particularly useful when functions have different left- and right-hand behaviors.

• A limit computed as \(x\) approaches from the left is denoted as \(x \to a^-\).
• When approaching from the right, it is written as \(x \to a^+\).

In our exercise, we are interested in a left-hand limit (\(x \rightarrow -4^-\)). When using one-sided limits, you often focus on re-evaluating the function's expression based on conditions like sign changes due to absolute values.
It allows us to define a limit when perhaps a standard two-sided limit doesn’t exist. For the exercise, we simplify \(\frac{|x+4|}{x+4}\) when \(x < -4\), resulting in \(-1\). This simplification leads to consistent results as \(x\) nears \(-4\) strictly from the left, ensuring the limit is clear.

Approaching from the Left

In calculus, understanding the direction of approach is important in determining limits, especially at points where the function definition might change.

• When we say "approaching from the left," we mean the values of \(x\) are getting closer to a specific point from lower values (i.e., smaller numbers).
• The notation for this is \(x \to a^-\).

For the problem at hand, as \(x\) approaches \(-4\) from the left, it entails examining values less than \(-4\).
This is significant since \(x < -4\) causes \(x+4\) to be negative, directly impacting the computation of \(|x+4|\) as \(-(x+4)\). It simplifies the expression to \(-1\), offering a constant value for the limit as \(x\) approaches \(-4\) from this direction. Directions like these define the pathway of approaching, causing sometimes different values to emerge compared to a broad approach from both sides, which underlines the utility of one-sided limits.