Problem 42 - Calculus: Early Transcendentals

Question

Find the limit, if it exists. If the limit does not exist, explain why. \( \displaystyle \lim_{x \to -6}\frac{2x + 12}{| x + 6 |} \)

Step-by-Step Solution

Answer

The limit does not exist because the left-hand and right-hand limits differ.

Step 1: Identify the critical point

First, we identify the critical point where the limit is evaluated: \( x = -6 \). At this point, the expression \(|x + 6|\) tends to zero. This indicates that the expression might be undefined unless we consider separate approaches as \(x\) approaches from different sides (left and right).

Step 2: Evaluate the left-hand limit

Evaluate the limit as \(x\) approaches \(-6\) from the left (denoted \(x \to -6^-\)). Here, \(x < -6\) and therefore \(|x + 6| = -(x + 6)\). Thus, the function becomes:\[ \frac{2x + 12}{-(x + 6)} = \frac{2(x + 6)}{-(x + 6)} = -2 \]The left-hand limit is \(-2\).

Step 3: Evaluate the right-hand limit

Evaluate the limit as \(x\) approaches \(-6\) from the right (denoted \(x \to -6^+\)). In this case, \(x > -6\) which implies \(|x + 6| = x + 6\). Thus, the function simplifies to:\[ \frac{2x + 12}{x + 6} = \frac{2(x + 6)}{x + 6} = 2 \]The right-hand limit is \(2\).

Step 4: Compare the left-hand and right-hand limits

Compare the results from Steps 2 and 3. The left-hand limit as \(x\) approaches \(-6\) is \(-2\), while the right-hand limit is \(2\). Since these two limits are not equal, the overall limit does not exist.

Key Concepts

Left-Hand Limit

The left-hand limit is all about understanding what happens to a function as it approaches a certain point from the left side. Here, we're talking about approaching from values that are smaller than the target point. Let's say you have a function, and you want to find the limit as the input variable approaches some critical value, say \(x = c\).

To find the left-hand limit, you would use \(x \to c^-\). This notation means that the function is being evaluated as \(x\) comes very close to \(c\) but from the left.

• The key feature is that these values are slightly less than \(c\).
• For piecewise functions, this might involve a specific expression that applies only when approaching from the left.

Since a left-hand limit focuses on values only approaching from one side, it often helps clarify the behavior of a function near edges, boundaries, or discontinuities.

Right-Hand Limit

The right-hand limit works similar to the left-hand limit but from the other side. With the right-hand limit, we look at what happens as the input variable of a function approaches a certain point from the right side, or from values that are slightly larger than the target point.

Here, the notation used is \(x \to c^+ \) which indicates approaching the critical value \(x = c \) from the right.

• This involves considering values just a bit greater than \(c\).
• In piecewise functions, there can be a separate rule or formula applied when approaching from the right.

If the right-hand limit matches the left-hand limit, the general point limit exists at that specific location. If they don't match, it means a discontinuity or jump exists.

Piecewise Functions

Piecewise functions are a fascinating part of mathematics and involve functions defined by multiple sub-functions, each with its own specific formula. These pieces are defined over various intervals of the input variable, like cutting a pizza into slices, where each slice has a different topping.

When working with piecewise functions, one of the critical aspects to consider is how each segment behaves around the points where the formula changes. This is where limits, especially left-hand and right-hand limits, come into play.

• Each section of a piecewise function may have different slopes, curvatures, and intercepts.
• Limit evaluations help determine continuity or discontinuity at the joining points.

Understanding piecewise functions can help explain unusual visual or physical phenomena that aren't represented well by a single mathematical expression.