Problem 56 - Calculus: Graphical, Numerical, Algebraic

Question

Group Activity In Exercises 55 and \(56,\) sketch a graph of a function \(y=f(x)\) that satisfies the stated conditions. Include any asymptotes. $$\begin{array}{l}{\lim _{x \rightarrow 2} f(x)=-1, \quad \lim _{x \rightarrow 4^{+}} f(x)=-\infty, \lim _{x \rightarrow 4^{-}} f(x)=\infty} \\ {\lim _{x \rightarrow \infty} f(x)=\infty, \lim _{x \rightarrow-\infty} f(x)=2}\end{array}$$

Step-by-Step Solution

Answer

The graph will start at \(f(x)=2\), decrease to -1 at \(x=2\), have a vertical asymptote at \(x=4\) where the function shoots up to infinity on the left and down to negative infinity on the right. For \(x\) values greater than 4, the function will rebound and rise indefinitely as \(x\) goes to infinity.

Step 1: Plot Given Limits

There are several different limiting behaviors given here. 1. As \(x\) approaches 2, \(f(x)\) approaches -1. 2. As \(x\) approaches 4 from the right, \(f(x)\) approaches negative infinity.3. As \(x\) approaches 4 from the left, \(f(x)\) approaches positive infinity.4. As \(x\) approaches positive infinity, \(f(x)\) approaches positive infinity.5. As \(x\) approaches negative infinity, \(f(x)\) approaches 2.For each limit, indicate the behavior on the graph. For the first limit, draw a point at \(x = 2, y = -1\). For the second and third, show that as \(x\) approaches 4, the function shoots up to positive infinity from the left and down to negative infinity from the right. Represent these behaviors with arrows. For the limits as \(x\) approaches infinity, show that as \(x\) gets large in the positive direction, \(f(x)\) goes up indefinitely, and as \(x\) gets large in the negative direction, \(f(x)\) approaches 2.

Step 2: Draw Asymptotes

Vertical asymptotes exist at discontinuities where the function goes to positive or negative infinity. In this case, there would be a vertical asymptote at \(x = 4\). Draw a dashed vertical line at \(x = 4\). Horizontal asymptotes represent the limiting behavior as the function goes to positive or negative infinity. From the given limits, there is no horizontal asymptote because the function does not approach a constant value as \(x\) goes to infinity or negative infinity.

Step 3: Construct the graph

Based on all the points and asymptotic behavior, draw a continuous line representing the function. This would involve starting from the leftmost part of the graph at \(y = 2\), decreasing towards \(y = -1\) at \(x = 2\), jumping to positive infinity and decreasing rapidly after \(x = 4\), and then rebounding and going to infinity as \(x\) goes to infinity.

Key Concepts

Limits

Limits are a fundamental concept in calculus that describe how a function behaves as the input, or variable, approaches a certain value. In simpler terms, a limit tells us what value a function is getting closer to as it gets closer to some point. Understanding limits is key to analyzing and drawing the graph of a function.

In the given exercise, we have several limits:

• As \(x\) approaches 2, the function \(f(x)\) approaches -1, meaning that near \(x=2\), \(f(x)\) gets closer to -1.
• As \(x\) approaches 4 from the right (denoted \(x \to 4^+\)), \(f(x)\) approaches negative infinity, indicating a sharp decrease.
• Conversely, as \(x\) approaches 4 from the left (\(x \to 4^-\)), \(f(x)\) heads towards positive infinity, showing a sharp increase.
• As \(x\) moves towards infinity, \(f(x)\) increases without bound.
• Lastly, as \(x\) tends towards negative infinity, \(f(x)\) stabilizes around 2.

Understanding these behaviors is crucial for sketching the graph accurately, as limits help predict the path the function takes as \(x\) approaches critical points.

Asymptotes

Asymptotes are lines that a graph approaches but never quite reaches. They provide a boundary that helps shape the behavior of functions, especially as variables move towards extreme values or certain points.

There are two types of asymptotes commonly discussed:

• Vertical Asymptotes: Occur where the function tends towards positive or negative infinity. In this exercise, there's a vertical asymptote at \(x = 4\), reflecting the behavior as \(f(x)\) diverges at this point.
• Horizontal Asymptotes: Suggest the direction a function will head as \(x\) moves towards positive or negative infinity. Here, the function lacks a horizontal asymptote because \(f(x)\) doesn't settle towards a constant as \(x\) becomes infinitely large in either direction.

Drawing asymptotes involves marking these boundaries on a graph with dashed lines, providing visual cues for how the function behaves near critical values.

Continuous Graph

A continuous graph of a function means that you can draw it without lifting your pencil from the paper. Continuity of a graph is important because it tells us that the function changes smoothly, without abrupt jumps or breaks.

In our exercise, the function should be continuous everywhere except at the vertical asymptote \(x = 4\). Here's how the continuity is demonstrated:

• The graph starts on the far left, approaching \(y = 2\) as \(x\) goes to negative infinity. This part of the graph should be smooth and constant.
• As the graph moves closer to \(x = 2\), it should glide towards \(-1\), then continue smoothly away again.
• Approaching \(x = 4\), the graph spikes to positive infinity from the left, then sharply descends to negative infinity from the right, causing the break at \(x = 4\).
• Finally, after passing \(x = 4\), the graph resumes continuity and rises ever upward as \(x\) heads to positive infinity.

Paying attention to continuity ensures the graph is plotted in a way that truthfully represents the function's nature, with the only discontinuity aligning precisely with the region of infinite behavior at the asymptote.