Problem 46 - Calculus: Early Transcendentals

Question

Sketching graphs of functions Sketch the graph of a function with the given properties. You do not need to find a formula for the function. $$f(1)=0, f(2)=4, f(3)=6, \lim _{x \rightarrow 2^{-}} f(x)=-3, \lim _{x \rightarrow 2^{+}} f(x)=5$$

Step-by-Step Solution

Answer

Answer: The function approaches -3 as a left-hand limit and 5 as a right-hand limit at x=2, showing a discontinuity (jump) at x=2.

Step 1: Plot the given points

First, let's plot the points given for x=1, x=2, and x=3. We have the following coordinates (1, 0), (2, 4), and (3, 6). Plot these points.

Step 2: Analyze the left-hand limit at x=2

The exercise asks for the left-hand limit (limit as x approaches 2 from the left) to be -3. This means that, as x approaches 2 from the left, the function should approach -3. So the graph of the function should go towards the y-coordinate of -3 as it moves close to x=2 from the left side.

Step 3: Analyze the right-hand limit at x=2

Similarly, the exercise asks for the right-hand limit (limit as x approaches 2 from the right) to be 5. As x approaches 2 from the right, the function should approach 5. So the graph of the function should go towards the y-coordinate of 5 as it moves close to x=2 from the right side.

Step 4: Sketch the graph around x = 2

Taking into consideration the given limits, sketch the graph around x=2 such that it approaches -3 on the left side of x=2 and it approaches 5 on the right side of x=2. The graph should have a discontinuity (jump) at x=2.

Step 5: Connect the remaining points

Now, connect the plotted points smoothly while adhering to the behavior of the graph around x=2. You should draw a continuous graph between the points (1, 0) and (3, 6), but with a jump (discontinuity) at x=2. You should now have a graph that satisfies the requirements given in the exercise.

Key Concepts

Left-Hand Limit

When we discuss the left-hand limit of a function, we're focusing on how the function behaves as the input value, denoted as \(x\), approaches a specific point from the left side. This is particularly useful when studying points of potential discontinuity or rapid change.
For the given example in the exercise, consider \( \lim _{x \rightarrow 2^{-}} f(x)=-3 \). This notation tells us that as \(x\) gets closer to 2 from values less than 2, the value of the function \(f(x)\) approaches -3.

• "Approaching from the left" means selecting values such as 1.9, 1.99, 1.999, etc., which are incrementally closer to 2 but still less than 2.
• Here, the function's curve should dip down towards a y-value of -3 as it gets near \(x=2\) from the left.

This insight allows us to understand a key aspect of the graph's behavior and prepare for any potential discontinuity or jump just before reaching \(x=2\). Understanding this helps ensure that the graph we sketch aligns perfectly with the conditions provided.

Right-Hand Limit

The concept of the right-hand limit mirrors that of the left-hand limit, but here we focus on approaching a point from values greater than the point of interest. This helps us understand how the function behaves as \(x\) approaches a certain value from the right side.
For instance, with \( \lim _{x \rightarrow 2^{+}} f(x)=5 \), we're instructed that as \(x\) approaches 2 from numbers like 2.1, 2.01, 2.001, and so on, the function \(f(x)\) heads towards 5.

• "Approaching from the right" involves selecting values greater than 2 but getting closer to 2, indicating the tendency or behavior of the graph's right approach.
• Therefore, on this side of the curve, the graph must shift towards a y-value of 5 as it nears \(x=2\).

Recognizing these tendencies is fundamental when sketching graphs because it contributes to accurately depicting how the graph changes and ensures it meets the condition specified for \(x=2\), completing the picture of how the function behaves on both sides around a specific point.

Discontinuity

Discontinuity in a graph tells us about sudden jumps, breaks, or gaps in the function's behavior. It plays a critical role when graphing functions with distinct left and right-hand limits at a point.
In our exercise scenario, the term discontinuity arises around \(x=2\) where the left-hand limit yields a value of -3, and the right-hand limit gives 5.

• This results in a "jump" discontinuity, where between the left and right approaches, the function breaks instead of remaining smooth or connected.
• At \(x=2\), instead of transitioning smoothly from one part to another, the graph rapidly jumps from one y-value to another.

A clear distinction within the graph can thus be visually discerned at this point, indicating that the curve splits, affirming the different values the function aims up to from each direction near \(x=2\). Properly understanding such discontinuities allows for more precise sketching of graphs and accurately represents the function's real behavior at points where the function does not seamlessly align.