Problem 40 - Calculus Early Transcendentals

Question

Find the limit, if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow-6} \frac{2 x+12}{|x+6|}$$

Step-by-Step Solution

Answer

The limit does not exist because the one-sided limits do not match.

Step 1: Identify Limit Approach

We need to evaluate the expression as \( x \) approaches -6. Notice the absolute value in the denominator, which should be carefully handled by considering the behavior of \( |x+6| \) for values just to the left and right of \( x = -6 \).

Step 2: Evaluate from the Left

As \( x \to -6^- \) (approaching -6 from the left), \( x+6 \) is negative. Thus, \(|x+6| = -(x+6)\). Substitute this in:\[\lim_{x \to -6^-} \frac{2x + 12}{-|x+6|} = \lim_{x \to -6^-} \frac{2x + 12}{-(x+6)} = \lim_{x \to -6^-} \frac{2x + 12}{-(x+6)}\] simplifying this to:\[\lim_{x \to -6^-} -2 = -2\] The limit from the left at \( x = -6 \) is -2.

Step 3: Evaluate from the Right

As \( x \to -6^+ \) (approaching -6 from the right), \( x+6 \) is positive. Thus, \(|x+6| = x+6\). Substitute this in:\[\lim_{x \to -6^+} \frac{2x + 12}{x+6} = \lim_{x \to -6^+} \frac{2x + 12}{x+6} = \lim_{x \to -6^+} 2 = 2\] The limit from the right at \( x = -6 \) is 2.

Step 4: Compare One-Sided Limits

The left-hand limit as \( x \to -6 \, \) is -2, while the right-hand limit as \( x \to -6 \, \) is 2. Since these one-sided limits are not equal, the two-sided limit does not exist.

Key Concepts

One-Sided Limits

When evaluating limits, especially when an absolute value or other discontinuities are involved, it is crucial to consider one-sided limits. These allow us to see how a function behaves as it approaches a particular point from two directions:

• From the left (denoted as \( x \to -6^- \))
• From the right (denoted as \( x \to -6^+ \))

By examining the one-sided limits separately, we can better understand if a two-sided limit exists. If the left-hand and right-hand limits are different, the overall limit at that point does not exist.
This is the case in the example we worked through, where the limit from the left was -2 and the limit from the right was 2. Since these one-sided limits did not match, the overall limit did not exist.

Absolute Value Function

The absolute value function, which is often written as \(|x|\), creates a situation where the behavior of a function can differ on each side of a particular point. Absolute value makes negative numbers positive but leaves positive numbers unchanged:

• If \( x \geq 0 \), then \(|x| = x\)
• If \( x < 0 \), then \(|x| = -x\) to achieve a positive result

For the problem we tackled, the function contained \(|x + 6|\). As \( x \) approaches -6, the behavior of \(|x + 6|\) changes.
That's why we had to break the problem down into evaluating the function coming from the left, where \( x + 6 \) becomes negative, and from the right, where \( x + 6 \) is positive. This allowed us to evaluate each side's limit with these considerations.

Limit Evaluation Methods

When evaluating limits, especially in cases involving absolute values or other peculiar function types, you might need various methods to reach a solution:

• Algebraic Manipulation: Simplifying expressions often makes it easier to evaluate limits.
• Piecewise Evaluation: For functions with absolute values, it helps to break the problem into parts and evaluate one-sided limits.
• Substitution: Directly substituting values to see results, but this needs to be used carefully, especially near undefined points.

In our example, we used piecewise evaluation. We considered the limit from both the left and right by setting \(|x + 6|\) to either \(x + 6\) or its negative form.
This strategy allowed us to evaluate the one-sided limits separately, using substitution in the simplified forms, and helped determine if the overall limit exists.