Problem 32 - Precalculus: Mathematics for Calculus

Question

One-Sided Limits Graph the piecewise-defined function and use your graph to find the values of the limits, if they exist. $$f(x)=\left\\{\begin{array}{ll} 2 x+10 & \text { if } x \leq-2 \\ -x+4 & \text { if } x>-2 \end{array}\right.$$ (a) \(\lim _{x \rightarrow-2^{-}} f(x)\) (b) \(\lim _{x \rightarrow-2^{+}} f(x)\) (c) \(\lim _{x \rightarrow-2} f(x)\)

Step-by-Step Solution

Answer

The limits are (a) 6, (b) 6, (c) 6.

Step 1: Understanding the Functions

The function is piecewise-defined with two sections: - For \( x \leq -2 \), the function is \( f(x) = 2x + 10 \).- For \( x > -2 \), the function is \( f(x) = -x + 4 \).

Step 2: Graphing the Left Function

Graph \( f(x) = 2x + 10 \) for \( x \leq -2 \). Calculate the endpoint:\[ x = -2: \quad f(-2) = 2(-2) + 10 = 6 \].As \( x \to -\infty \), the graph follows the line \( f(x) = 2x + 10 \). Mark the point \((-2, 6)\) as solid since it's defined in this piece.

Step 3: Graphing the Right Function

Graph \( f(x) = -x + 4 \) for \( x > -2 \). Find a starting point:\[ x = -2: \quad f(-2)(approach from right) = -(-2) + 4 = 6 \].This point \((-2, 6)\) is not part of this graph, so mark it on the graph with an open circle to distinguish it from the defined left piece.

Step 4: Evaluating the Left-Sided Limit

Evaluate \( \lim_{x \to -2^-} f(x) \) using the left segment \( f(x) = 2x + 10 \):Calculate: \[ \lim_{x \to -2^-} (2x + 10) = 2(-2) + 10 = 6 \].Therefore, \( \lim_{x \to -2^-} f(x) = 6 \).

Step 5: Evaluating the Right-Sided Limit

Evaluate \( \lim_{x \to -2^+} f(x) \) using the right segment \( f(x) = -x + 4 \):Calculate: \[ \lim_{x \to -2^+} (-x + 4) = 2 + 4 = 6 \].Thus, \( \lim_{x \to -2^+} f(x) = 6 \).

Step 6: Evaluating the Two-Sided Limit

Check if both one-sided limits are equal. Since \( \lim_{x \to -2^-} f(x) = 6 \) and \( \lim_{x \to -2^+} f(x) = 6 \),\[ \lim_{x \to -2} f(x) = 6 \]. The two-sided limit exists and is 6.

Key Concepts

Piecewise-Defined Function

A piecewise-defined function is composed of different expressions based on different parts of the domain. It means that the function behaves differently depending on the interval of the input values. For example, in the given exercise, the function is defined by two expressions:

• For inputs where \( x \leq -2 \), the function is \( f(x) = 2x + 10 \).
• For inputs where \( x > -2 \), the function is \( f(x) = -x + 4 \).

The key idea is that you are tentatively switching between these expressions based on the value of \( x \). When dealing with piecewise functions, it's critical to pay attention to the inequalities that define each section. This is what guides you in determining how the function behaves over various parts of its domain.

Graphing Functions

Graphing a piecewise-defined function involves plotting each segment separately on a coordinate plane and paying attention to the domains they cover. When graphing:

• Determine the equations used for each piece and the intervals over which they apply.
• Plot only the relevant segment of the graph according to its interval, using solid or open dots to indicate whether endpoints are included.

For the given example:- For \( x \leq -2 \), graph \( f(x) = 2x + 10 \) ensuring a solid dot at \((-2, 6)\) as it includes \( x = -2 \).
- For \( x > -2 \), graph \( f(x) = -x + 4 \) starting just after \( x = -2 \) and using an open dot to show that the function does not include \( x = -2 \).
Graphing accurately is crucial as it visually illustrates how functions behave across different intervals of the domain.

Evaluating Limits

Evaluating limits involves determining what value a function approaches as the input gets infinitely close to a particular point. Sometimes, this involves checking from both sides (left and right) to get a complete picture. Here's how it's done:

• For the left-sided limit \( \lim_{x \to -2^-} f(x) \), you consider only the values approaching \( -2 \) from the left (i.e., \( x \leq -2 \)).
• For the right-sided limit \( \lim_{x \to -2^+} f(x) \), use the function values moving towards \( -2 \) from the right (i.e., \( x > -2 \)).

In both cases, it's essential to substitute values close to \( x = -2 \) into the respective function expressions. If the results from both sides match, it indicates that the overall limit at that point is the same and exists, confirming continuity at that point.

Two-Sided Limit

A two-sided limit at a point occurs when the left-sided limit and the right-sided limit at that point are equal. In the exercise, this is verified by comparing the results from evaluating limits at \( x = -2 \) from both sides:

• The left-sided limit, \( \lim_{x \to -2^-} f(x) \), is \( 6 \).
• The right-sided limit, \( \lim_{x \to -2^+} f(x) \), is \( 6 \) too.

Since both are equal, \( \lim_{x \to -2} f(x) \) is \( 6 \), and hence, the two-sided limit exists. Identifying two-sided limits helps confirm function continuity at specific points and ensures that the function doesn't have jumps or breaks at those points.