Problem 9 - Calculus: Early Transcendentals

Question

Sketch the graph of an example of a function \( f \) that satisfies all of the given conditions. \( f(0) = 3 \), \( \displaystyle \lim_{x \to 0^-} f(x) = 4 \), \( \displaystyle \lim_{x \to 0^+} f(x) = 2 \), \( \displaystyle \lim_{x \to -\infty} f(x) = -\infty \), \( \displaystyle \lim_{x \to 4^-} f(x) = -\infty \), \( \displaystyle \lim_{x \to 4^+} f(x) = \infty \), \( \displaystyle \lim_{x \to \infty} f(x) = 3 \)

Step-by-Step Solution

Answer

Sketch a discontinuous graph with correct behavior at each specified point and limit.

Step 1: Identify Given Conditions

Identify all the conditions given in the problem:- Discontinuity at \( x = 0 \) where \( f(0) = 3 \), but limits are different as \( \lim_{{x \to 0^-}} f(x) = 4 \) and \( \lim_{{x \to 0^+}} f(x) = 2 \).- End behavior as \( x \to -\infty: \lim_{x \to -\infty} f(x) = -\infty \).- Discontinuity at \( x = 4 \) with limits \( \lim_{{x \to 4^-}} f(x) = -\infty \) and \( \lim_{{x \to 4^+}} f(x) = \infty \).- End behavior as \( x \to \infty: \lim_{x \to \infty} f(x) = 3 \).

Step 2: Sketch the Behavior at Each Discontinuity

For \( x = 0 \), plot a point at (0, 3). The left-hand limit approaches 4, so sketch the curve as approaching the line \( y = 4 \) from the left. The right-hand limit approaches 2, so sketch it approaching \( y = 2 \) from the right. Leave a gap at \( x = 0 \) indicating a jump discontinuity.

Step 3: Sketch the Behavior at Infinity Points

As \( x \to -\infty \), the function goes to \( -\infty \). Sketch a plot that starts very low on the left side of the graph to indicate this behavior. As \( x \to \infty \), \( f(x) \to 3 \). Draw the graph leveling out to approach \( y = 3 \) from the left, without intersecting.

Step 4: Sketch the Behavior Around x=4

At \( x = 4 \), the left limit approaches \( -\infty \) and the right limit approaches \( \infty \), indicating a vertical asymptote at \( x = 4 \). Sketch a curve that drops down towards \( -\infty \) as it approaches from the left, and rises towards \( \infty \) from the right.

Step 5: Connect the Graph Sections

Link all the sections smoothly, maintaining the specified behaviors at the boundaries. Ensure that gaps exist at the jump discontinuities and the infinite limits are clearly represented to complete the graph. Confirm that all conditions are met.

Key Concepts

Understanding Discontinuity

Discontinuity occurs when a function is not continuous at a particular point. This means that the function does not have a single, unbroken path at this location. In our exercise, we observe two discontinuities at points \( x = 0 \) and \( x = 4 \).

At \( x = 0 \), the function value is given as \( f(0) = 3 \). However, the left-hand limit, \( \lim_{x \to 0^-} f(x) \), approaches 4 and the right-hand limit, \( \lim_{x \to 0^+} f(x) \), approaches 2. This results in a jump discontinuity because the different values on either side indicate a sudden change or 'jump' in the value of the function as you pass through \( x = 0 \).

For \( x = 4 \), the function does not have a defined value but it shows another type of discontinuity known as an infinite discontinuity. Here, the limits from either side approach infinite values: \( \lim_{x \to 4^-} f(x) = -\infty \) and \( \lim_{x \to 4^+} f(x) = \infty \). This infinite or vertical asymptote marks that as \( x \) gets very close to 4, the values of the function increase or decrease without bound.

Exploring Limits of Functions

The concept of limits is fundamental in calculus and deals with the behavior of a function as it approaches a particular point from either direction. Limits provide crucial information about the function, especially in discontinuous and continuous ranges.

In our function exploration, we focus on several limits:

• \( \lim_{x \to 0^-} f(x) = 4 \) and \( \lim_{x \to 0^+} f(x) = 2 \)
• \( \lim_{x \to -\infty} f(x) = -\infty \)
• \( \lim_{x \to \infty} f(x) = 3 \)
• \( \lim_{x \to 4^-} f(x) = -\infty \) and \( \lim_{x \to 4^+} f(x) = \infty \)

These limits indicate how the function behaves near and far from certain points. At \( x = 0 \), the differing limits from left and right show a jump in the function. At infinite approaches like \( x \to -\infty \), the function tends towards \( -\infty \), illustrating its downward trend as \( x \) becomes very large in the negative direction. Similarly, as \( x \) moves positively to infinity, \( f(x) \) stabilizes towards 3, showing a horizontal asymptote.

Understanding Asymptotic Behavior

Asymptotic behavior describes how a function behaves as it moves towards certain values or infinity. This behavior is key in visualizing the function approaching a line or curve but never actually reaching it, be it a horizontal or vertical asymptote.

In the given function, we see the presence of both vertical and horizontal asymptotes:

• Vertical asymptote at \( x = 4 \), where the function goes to \( -\infty \) from the left and \( \infty \) from the right.
• Horizontal asymptote as \( x \to \infty \) with \( f(x) \to 3 \), suggesting that the function flattens out toward this value.

Vertical asymptotes occur where a function's limit does not exist because it shoots off to infinity, either positive or negative. Horizontal asymptotes indicate that as \( x \) becomes very large (positively or negatively), the function's values approach a constant value. Understanding these limits helps predict the function's long-term behavior, crucial for sketching and analyzing the function accurately.