Problem 63 - Brief Applied Calculus

Question

For each piecewise linear function: a. Draw its graph (by hand or using a graphing calculator). b. Find the limits as \(x\) approaches 3 from the left and from the right. c. Is it continuous at \(x=3\) ? If not, indicate the first of the three conditions in the definition of continuity (page 87) that is violated. $$ f(x)=\left\\{\begin{array}{ll} x & \text { if } x \leq 3 \\ 7-x & \text { if } x>3 \end{array}\right. $$

Step-by-Step Solution

Answer

The function is not continuous at \( x = 3 \) because the left-hand and right-hand limits are not equal.

Step 1: Understanding the Piecewise Function

The given function is a piecewise function defined as follows: \( f(x) = x \) if \( x \leq 3 \), and \( f(x) = 7 - x \) if \( x > 3 \). Our job is to analyze this function, draw its graph, and investigate its continuity at \( x = 3 \).

Step 2: Drawing the Graph

To draw the graph, we need to plot the line \( y = x \) for \( x \leq 3 \), which is a straight line passing through the origin with a slope of 1, continuing up to the point (3, 3). For \( x > 3 \), we need to plot \( y = 7 - x \), which is a line with a negative slope where it intercepts the y-axis at 7. This line starts immediately to the right of \( x = 3 \) at (3, 4) and continues downwards.

Step 3: Finding the Limit from the Left

To find the limit as \( x \to 3^- \), we consider the function \( f(x) = x \). As \( x \) approaches 3 from the left, the function value approaches 3. Thus, \( \lim_{x \to 3^-} f(x) = 3 \).

Step 4: Finding the Limit from the Right

To find the limit as \( x \to 3^+ \), consider the function \( f(x) = 7 - x \). As \( x \) approaches 3 from the right, the function value approaches \( 7 - 3 = 4 \). Thus, \( \lim_{x \to 3^+} f(x) = 4 \).

Step 5: Determining Continuity

A function is continuous at a point if the left-hand limit, right-hand limit, and the function value at that point are all equal. In this case, \( \lim_{x \to 3^-} f(x) = 3 \) and \( \lim_{x \to 3^+} f(x) = 4 \). Since the left and right limits are not equal, \( f(x) \) is not continuous at \( x = 3 \). The first condition violated is that the left-hand and right-hand limits must be equal for continuity.

Key Concepts

Understanding Limits in Piecewise Functions

In calculus, the concept of limits helps us understand how functions behave as they approach a specific point from both sides. For piecewise functions, such as the one given in the exercise, finding limits is crucial.
The function is defined differently on either side of the point of interest, in this case, \( x = 3 \).
To determine a limit, take the values of \( x \) as they get closer and closer to the point you’re interested in from either direction.
You can visualize this by considering what happens to \( f(x) \) as \( x \) approaches 3 from the left and right. The limit from the left is not necessarily equal to the limit from the right, which can affect the continuity of the function.

Graphing Piecewise Functions

Graphing is a powerful way to visualize how piecewise functions behave.
For this exercise, the function is defined by different expressions depending on whether \( x \) is less than or greater than 3.
- For \( x \leq 3 \), the function follows \( y = x \), a simple line moving upward from the origin.- For \( x > 3 \), it switches to \( y = 7 - x \), forming a line declining from \( y = 4 \) at the point \( (3, 4) \).
Using graph paper or software to plot these can help you easily see where the function jumps or changes, aiding in checking continuity and limit values.

Exploring Continuity at a Point

Continuity means you can draw a function without lifting your pencil from the paper.
A function is continuous at a specific point if:

• The limit from the left and right at that point are equal.
• The value of the function at that point is defined and equals these limits.

With our piecewise function, we observe that \( \lim_{x \to 3^-} f(x) = 3 \) while \( \lim_{x \to 3^+} f(x) = 4 \). Since these are not equal, \( f(x) \) is not continuous at \( x = 3 \). This lack of equivalence violates the first condition of continuity.

Understanding the Left-Hand Limit

The left-hand limit is what the function value approaches as \( x \) comes from values smaller than the point of interest.
For our function, as \( x \to 3^- \), we look at \( f(x)=x \) for values just less than 3.
As \( x \) inches closer to 3 from below, \( f(x) \) approaches 3, i.e., \( \lim_{x \to 3^-} f(x) = 3 \).
Thinking of the graph, it's like asking: "If I come along this part of the graph, where does it aim right before I get to 3?"

Understanding the Right-Hand Limit

Conversely, the right-hand limit focuses on approaching from values greater than the point.
For \( x \to 3^+ \), we use the function part defined for \( x > 3 \), which is \( f(x)=7-x \).
As you approach closer to \( x = 3 \) from this direction, \( f(x) \) attempts to reach 4: \( \lim_{x \to 3^+} f(x) = 4 \).
On the graph, this would mean examining the direction you're heading right after passing 3 from the larger side.