Problem 6 - Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The function is hyperbolic, with vertical asymptote at \( x = 2 \) and horizontal asymptote at \( y = 0 \).

Step 1: Interpret the Condition at x=2

The condition \( \lim_{x \to 2} f(x) = \infty \) means the function \( f(x) \) approaches infinity as \( x \) gets closer to 2. Additionally, \( \lim_{x \to 2^+} f(x) = \infty \) implies from the right side of 2, \( f(x) \) goes to infinity. However, \( \lim_{x \to 2^-} f(x) = -\infty \) indicates that approaching 2 from the left, \( f(x) \) goes to negative infinity. This is typical for a vertical asymptote at \( x = 2 \) where the function diverges differently from each side.

Step 2: Analyze the End Behavior

The conditions \( \lim_{x \to -\infty} f(x) = 0 \) and \( \lim_{x \to \infty} f(x) = 0 \) suggest that as \( x \) moves towards both negative and positive infinity, the function \( f(x) \) approaches 0. This indicates that the function has horizontal asymptotes at \( y = 0 \).

Step 3: Use f(0)=0 Condition

The point \( f(0) = 0 \) means the function passes through the origin (0,0). This provides a specific point that must be included in the graph to satisfy the given conditions.

Step 4: Sketch the Function

Begin by drawing a vertical asymptote at \( x = 2 \). On the right side of 2, the function should rise towards infinity, and on the left side, it should fall towards negative infinity. Next, draw the horizontal asymptote \( y = 0 \) extended from both negative infinity to positive infinity. Make sure the function curves upward and passes through the origin \( (0, 0) \). Combine these behaviors into a coherent sketch of \( f(x) \), resembling a hyperbola rotated, showing the specified habits around the asymptotes and zero point.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur in a function where the output value tends to infinity as the input variable, often denoted as \( x \), approaches a certain point. This happens because the function is not defined at that particular input value, leading the function to shoot up to positive or negative infinity. In the original exercise, we observe vertical asymptotes at \( x = 2 \). Here's what's happening:

• As \( x \) approaches 2 from the right (symbolized by \( x \to 2^+ \)), \( f(x) \) heads towards positive infinity.
• From the left (symbolized by \( x \to 2^- \)), \( f(x) \) plummets towards negative infinity.

These behaviors at \( x = 2 \) depict a typical vertical asymptote, creating a gap or a spike in the graph.
The distinctive behavior of diverging from different sides often indicates a rational function where the denominator equals zero at that input value.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the input variable approaches very large positive or negative numbers. They are horizontal lines that the graph of the function approaches but does not necessarily touch or cross, as \( x \to \pm \infty \).
In our exercise, we see:

• As \( x \to -\infty \), \( f(x) \to 0 \).
• As \( x \to \infty \), \( f(x) \to 0 \).

These conditions show that the graph approaches the line \( y = 0 \) on each side, implying that the function has a horizontal asymptote at \( y = 0 \).Horizontal asymptotes are significant in helping to predict the long-term behavior of functions. They tell us what the function's values tend towards but don't provide exact specifics of the function's behavior; for instance, functions can cross horizontal asymptotes at certain points.

Behavior of Functions at Infinity

The behavior of functions as \( x \) increases or decreases towards infinity is crucial for understanding trends and potential outcomes of a function's graph over a broad range.
In the original problem, the limits at infinity are given:

• \( \lim_{x \to -\infty} f(x) = 0 \)
• \( \lim_{x \to \infty} f(x) = 0 \)

This states that as \( x \) moves towards both extremes (negative and positive infinity), the function \( f(x) \) approaches 0.
This type of behavior means the function becomes very close to the \( x \)-axis in both directions, resembling a flattening effect as the input grows large or very small. It's essential to note that the factors such as the degree of polynomials in rational functions or exponential components play massive roles in dictating this behavior.Understanding these behaviors assists in forming accurate sketches of the function and anticipating how the function acts beyond what's immediately visible in a graph.