Problem 37 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Problem

We need to find the limit of the absolute value function \(|x + 4|\) as \(x\) approaches \(-4\). The absolute value function \(|x + 4|\) represents how far \(x + 4\) is from zero on the number line, regardless of direction.

Step 2: Evaluate the Absolute Value Function at the Limit Point

Substitute \(x = -4\) into the expression to find \(|-4 + 4|\). This becomes \(|0| = 0\).

Step 3: Consider the Left-Hand and Right-Hand Limits

Calculate the limits as \(x\) approaches \(-4\) from the left and the right.When \(x > -4\), \(x + 4\) is slightly positive, so \(|x + 4| = x + 4\).When \(x < -4\), \(x + 4\) is slightly negative, so \(|x + 4| = -(x + 4)\).However, regardless of approaching from the left or right, as \(x\) gets closer to \(-4\), both forms \(|x + 4|\) evaluate to 0 as \(x + 4 \to 0\).

Step 4: Conclusion

Both the left-hand limit and right-hand limit converge to 0. Therefore, the overall limit exists and is 0.

Key Concepts

Absolute Value

The absolute value of a number simply tells us how far the number is from zero on the number line, without considering any direction or sign. It is denoted by two vertical bars around the number or expression, like \(|x+4|\).
So, for an expression like \(x+4\), the absolute value \(|x+4|\) is always non-negative, because it represents a distance. No matter if \(x+4\) is positive or negative itself, \(|x+4|\) will yield the same non-negative number.
When we look for the limit of an absolute value function, like \(|x+4|\) in this case, we consider what happens to its value as \(x\) gets close to a certain point. These ideas help clarify the meaning and computation of limits involving absolute values.

One-Sided Limits

One-sided limits help us analyze what happens to a function as the input approaches a specific value from only one direction. This is particularly useful when the function behaves differently on either side of the point we are examining.
For the exercise involving \(|x + 4|\), when calculating as \(x\) approaches \(-4\), we consider:

• The right-hand limit: approaching \(-4\) from values greater than \(-4\) (meaning \(x > -4\))
• The left-hand limit: approaching \(-4\) from values less than \(-4\) (so \(x < -4\))

For the function \(|x + 4|\), both left-hand and right-hand limits evaluate to zero as \(x + 4 o 0\). Therefore, the function does not approach different values when coming from different directions, allowing us to find a common limit.

Limit Evaluation

Evaluating a limit means determining what value a function approaches as its input approaches a certain number. For most simple functions, substitution can be used, but more complex functions might require different tactics.
In our exercise, when evaluating \(\lim_{x \to -4} |x+4|\), focus on computing the absolute value carefully as \(x\) nears \(-4\). By evaluating both one-sided limits, we found that the absolute value expression simplifies directly to zero. This is straightforward once we recognize the behavior of absolute value near the point \(-4\).
By considering both \(x < -4\) and \(x > -4\), and their respective impacts on the absolute value, we conclude that the limit indeed exists as both sides converge to the same value, in this case, zero.

Convergence

Convergence in limits means that as \(x\) gets closer to a specific point, the function approaches a single definite value. For convergence to be established in evaluating a limit, both the left-hand and right-hand limits must be equal.
In the context of our exercise, as \(x\) approaches \(-4\), both one-sided computations show that \(|x+4|\) compresses towards zero. Thus, we assert that the limit exists and converges to zero.
When we can clearly establish that a function converges to the same value from different approaches, we gain confidence in our limit's accuracy and correctness, ensuring the concept underlying our solution reflects mathematical integrity.