Problem 38 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The limit is -1.

Step 1: Understand the Expression

The given expression is \( \lim _{x \rightarrow-4^{-}} \frac{|x+4|}{x+4} \). We need to find the limit as \( x \) approaches -4 from the left. This means we consider values of \( x \) that are just less than -4.

Step 2: Simplify the Absolute Value

For values of \( x \) less than -4, the expression inside the absolute value \( |x+4| \) becomes negative because \( x+4 < 0 \). Therefore, \( |x+4| = -(x+4) \).

Step 3: Substitute and Simplify

Substitute \( |x+4| = -(x+4) \) into the expression: \[ \frac{|x+4|}{x+4} = \frac{-(x+4)}{x+4} \]. Simplifying this gives us \( \frac{-(x+4)}{x+4} = -1 \).

Step 4: Evaluate the Limit

As \( x \to -4^{-} \), the expression simplifies to -1. Therefore, the limit of \( \frac{|x+4|}{x+4} \) as \( x \to -4^{-} \) is \( -1 \).

Key Concepts

Understanding Absolute Value

The concept of absolute value is essential in understanding limits, especially when dealing with expressions that involve inequalities. Absolute value, denoted as \(|x|\), represents the distance of a number from zero on the number line. Consequently, it is always a non-negative number. When looking at \(|x + 4|\), you should consider two cases:

• If \(x + 4\) is positive or zero, then \(|x + 4| = x + 4\).
• If \(x + 4\) is negative, then \(|x + 4| = -(x + 4)\).

In the context of the limit \(\lim_{x \rightarrow -4^{-}} \dfrac{|x+4|}{x+4}\), notice that we are dealing with the second case because we examine values just less than \(-4\), meaning \(x + 4\) is indeed negative. Thus, knowing how absolute value behaves is crucial for simplifying such expressions.

Exploring One-Sided Limits

One-sided limits are significant when investigating the behavior of functions as the input approaches a specific point from one direction. Specifically, a one-sided limit at \(x \rightarrow c^{-}\) considers values of \(x\) that are slightly less than \(c\), whereas a limit of \(x \rightarrow c^{+}\) would consider values slightly greater. In this exercise, we are asked to find \(\lim_{x \rightarrow -4^{-}} \dfrac{|x+4|}{x+4}\). Here, the expression needs examination only from the left-hand side because it's a left-hand limit (denoted by \(-\)). Staying strictly on the left hand side of \(-4\) ensures that the nature of the fraction \(|x + 4|/(x + 4)\) remains consistent, which is vital in accurately identifying the limit value without ambiguity.One-sided limits help illuminate the behavior of expressions at boundary points and are quite useful when dealing with piecewise and discontinuous functions, like the one observed here with absolute values.

Simplifying Expressions for Limits

Simplifying expressions is an essential skill for solving limit problems, especially when involving complicated operations like division or absolute values. Simplification often involves algebraic manipulation to reveal the core structure of an expression.In the case of \(\frac{|x+4|}{x+4}\), simplification happens by addressing the absolute value. As established, when \(x < -4\), \(|x+4|\) simplifies to \(-(x+4)\). Therefore, the fraction simplifies to:

• \(\frac{-(x+4)}{x+4} = -1\).

After simplifying, determining the limit becomes straightforward since the expression has resolved to a constant. Simplifying expressions correctly ensures that the limiting behavior reflects the true nature of the expression without the risk of division by zero or other such pitfalls. Always simplify the algebraic expression as much as possible to see if any insights emerge, which might help in establishing the limit easily.