Problem 10 - Calculus: Early Transcendentals

Question

Sketch the graph of an example of a function \( f \) that satisfies all of the given conditions. \( \displaystyle \lim_{x \to 3} f(x) = -\infty \), \( \displaystyle \lim_{x \to \infty} f(x) = 2 \), \( f(0) = 0 \), \( f \) is even

Step-by-Step Solution

Answer

A function like \( f(x) = \frac{1}{(x^2 - 9)} + 2 \) satisfies the conditions.

Step 1: Analyze the Conditions

We have four conditions for the function \( f \):1. \( \lim_{x \to 3} f(x) = -\infty \): This indicates a vertical asymptote at \( x = 3 \).2. \( \lim_{x \to \infty} f(x) = 2 \): The function approaches a horizontal asymptote at \( y = 2 \) as \( x \to \infty \).3. \( f(0) = 0 \): The function passes through the point (0,0).4. \( f \) is even: This means \( f(x) = f(-x) \) for all \( x \). Thus, the function is symmetric about the y-axis.

Step 2: Sketch the Asymptotes

Draw the vertical asymptote at \( x = 3 \) and a horizontal asymptote at \( y = 2 \) on the graph. These will guide the behavior of the function.

Step 3: Place the Known Points

Plot the point (0,0) on the graph since \( f(0) = 0 \). Ensure the graph is symmetric about the y-axis due to the even function property.

Step 4: Determine the Behavior around Asymptotes

The function should dramatically decrease as it approaches \( x = 3 \) from both sides, going down to \(-\infty\). For an even function, this behavior should reflect symmetrically about the y-axis.

Step 5: Define Behavior Towards Infinity

As \( x \to \infty \), the function should approach \( y = 2 \) from below or above, depending on the function's specific construction. Because \( f \) is even, as \( x \to -\infty \), the function should also approach \( y = 2 \).

Step 6: Sketch the Graph

Connect the known points and asymptotic behavior in a smooth, continuous curve. Start from \( x = 0 \), going to negative infinity, approach \(-\infty\) vertically at \( x = 3 \), and ascend towards \( y = 2 \) as \( x \to \infty \). Ensure the curve is symmetrical about the y-axis.

Key Concepts

Vertical Asymptote

When analyzing functions, one important feature is the vertical asymptote. A vertical asymptote occurs when a function becomes infinitely large or infinitely small as it approaches a specific value of the independent variable, usually denoted as \( x \). For the function in our exercise, the condition \( \lim_{x \to 3} f(x) = -\infty \) tells us there is a vertical asymptote at \( x = 3 \). As you near this value, the function's values head toward negative infinity.

• To represent this behavior graphically, we draw a dotted vertical line on the graph at \( x = 3 \).
• This line indicates to us that the function will not cross or touch \( x = 3 \).
• The graph will instead plummet downwards dramatically as it gets closer to this vertical line, representing the blow-up to negative infinity.

Understanding vertical asymptotes is crucial for predicting the behavior of functions as inputs approach certain values.

Horizontal Asymptote

Horizontal asymptotes show the behavior of a function as the input grows very large (positively or negatively). It describes what value the function will approach but never reach. For our function, we see that \( \lim_{x \to \infty} f(x) = 2 \). This tells us there is a horizontal asymptote at \( y = 2 \).

• This means, as you extend the graph far to the right (where \( x \) values are very large), the function approaches the value 2.
• Importantly, for even functions like ours, this behavior also applies symmetrically when \( x \to -\infty \), so the graph also approaches \( y = 2 \) on the far left.
• A horizontal line at \( y = 2 \) should be drawn to represent this asymptote.

Horizontal asymptotes help us grasp the end behavior of functions, letting us know the value they are trying to settle at over time.

Even Function

An even function is characterized by its symmetry about the y-axis. In practical terms, this means that for any given value of \( x \), \( f(x) = f(-x) \). This rule holds true for all \( x \) in the function's domain.

• Because the function \( f \) is even, whatever happens on the right of the y-axis also happens on the left.
• The point (0,0) mentioned in the conditions highlights this symmetry start. Plotting the graph, keep this point as a central hinge, so the graph looks the same in both directions away from the y-axis.
• This symmetry must reflect in how the curve behaves as it interacts with its vertical and horizontal asymptotes.

Learning about even functions can guide the drawing and analysis of function graphs by predicting how they mirror across the y-axis.

Limit Behavior

Limit behavior is the cornerstone in understanding how a function behaves as it nears specific points, both finite and infinite.The limits provided in the conditions give insight into the behavior at particular critical points and at infinity.

• For the limit \( \lim_{x \to 3} f(x) = -\infty \), we see how \( f(x) \) exhibits a vertical asymptotic behavior, forcing the function to descend indefinitely as \( x \) approaches 3.
• With \( \lim_{x \to \infty} f(x) = 2 \), the graph gradually nears the horizontal asymptote at \( y = 2 \), stabilizing as \( x \) gets large.
• Similarly, because our function is even, \( \lim_{x \to -\infty} f(x) = 2 \) as well, making sure that as \( x \) moves towards negative infinity, the function still approaches the same value \( y = 2 \).

Understanding limit behavior is key to anticipating the course of a function's graph and effectively drawing its path.