Problem 5 - Calculus: Early Transcendentals

Question

Sketch the graph of an example of a function \( f \) that satisfies all of the given conditions. \( \displaystyle \lim_{x \to 0} f(x) = -\infty \), \( \displaystyle \lim_{x \to -\infty} f(x) = 5 \), \( \displaystyle \lim_{x \to \infty} f(x) = -5 \)

Step-by-Step Solution

Answer

The function \(f(x) = \frac{10x}{x^2-1}\) satisfies all given conditions.

Step 1: Understanding the Limits at Infinity

The problem gives us three conditions about the behavior of the function at different points. First, consider the limits as \(x\) approaches negative and positive infinity. \( \lim_{x \to -\infty} f(x) = 5 \) tells us that as \(x\) moves towards negative infinity, \(f(x)\) approaches 5. Similarly, \( \lim_{x \to \infty} f(x) = -5 \) means as \(x\) moves towards positive infinity, \(f(x)\) approaches -5. This suggests horizontal asymptotes at \(y = 5\) and \(y = -5\) as \(x\) approaches negative and positive infinity, respectively.

Step 2: Understanding the Limit at Zero

Next, interpret \( \lim_{x \to 0} f(x) = -\infty \). This means that as \(x\) approaches 0, the function \(f(x)\) goes to negative infinity. This suggests a vertical asymptote at \(x = 0\), indicating that the function diverges negatively as you approach \(x = 0\) from either side.

Step 3: Choosing a Function Based on the Conditions

Based on these conditions, a rational function might be a good candidate because it can have both vertical and horizontal asymptotes. Consider the function \( f(x) = \frac{10x}{x^2-1} \). This function approaches 5 as \(x\) approaches negative infinity, -5 as \(x\) approaches positive infinity, and has a vertical asymptote at \(x = 0\) making it go to negative infinity.

Step 4: Verifying the Chosen Function Meets All Conditions

Verify the function \( f(x) = \frac{10x}{x^2-1} \) against each condition. As \(x\) approaches 0, the denominator \(x^2 - 1\) approaches \(-1\), but the numerator remains \(10x\). Thus, \( \lim_{x \to 0} f(x) = -\infty \) as \(x\) approaches 0. For the horizontal asymptotes, for large \(|x|\), the function simplifies to \(\frac{10x}{x^2}\) which simplies to \(\frac{10}{x}\), leading to \(5\) and \(-5\) as \(x\) approaches negative and positive infinity, respectively.

Step 5: Sketching the Graph

Plot the graph with x-axis and y-axis. Draw horizontal asymptotes at \(y = 5\) and \(y = -5\). Draw a vertical asymptote at \(x = 0\). Sketch the function \(f(x) = \frac{10x}{x^2-1}\) so that it approaches these asymptotes appropriately as described in previous steps.

Key Concepts

Asymptotes

An asymptote is a line that a graph of a function approaches without actually touching it. They play a crucial role in understanding the behavior of functions, especially at the boundaries of the region of interest. Asymptotes can be vertical, horizontal, or even slanted.

- **Vertical Asymptotes** occur where a function's value heads towards infinity as the input approaches a particular value. For instance, if a function has a vertical asymptote at \(x=0\), then as \(x\) gets close to 0 from either direction, the function's value will shoot up or down toward infinity. This is seen in the function \( f(x) = \frac{10x}{x^2-1} \) as \(x\) approaches 0 with \(f(x)\) going to \(-\infty\).
- **Horizontal Asymptotes** describe the end behavior of a function, showing how it behaves as \(x\) approaches positive or negative infinity. A function like \( f(x) = \frac{10x}{x^2-1} \) has horizontal asymptotes at \(y=5\) and \(y=-5\). This indicates that as \(x\) goes to negative infinity, \(f(x)\) tags along near 5, and as \(x\) goes to positive infinity, \(f(x)\) approaches -5.

Understanding asymptotes and their implications can help identify behaviors such as growth patterns, end behaviors, and potential discontinuities in functions.

Rational Functions

Rational functions are like the bread and butter of calculus and algebra. These functions are ratios of two polynomials, like \( f(x) = \frac{10x}{x^2-1} \) where you see a polynomial divided by another polynomial.

Here’s what makes them special:

• They can have vertical and horizontal asymptotes, making them super useful for seeing extreme behavior of functions as \(x\) heads towards infinity or particular points.
• Their domains are all real numbers except where the denominator equals zero. For example, in \( f(x) = \frac{10x}{x^2-1} \), the values that make \(x^2-1=0\) (which happens when \(x^2=1\) or \(x=\pm1\)) are not in the domain.
• When analyzing or sketching graphs, identifying horizontal asymptotes tells us about the end behavior of the graph, while vertical asymptotes can shed light on where the function might head to infinity.

Rational functions provide a flexible framework to model real-world situations where relationships between quantities are not straightforward but are influenced by different degrees of influence.

Graph Sketching in Calculus

Graph sketching in calculus helps visualize how a function behaves across different intervals and as it approaches certain limits. Let's break down the process using the function from above, \( f(x) = \frac{10x}{x^2-1} \).

**Steps for Sketching a Graph:**

• First, identify the asymptotes. For this function, there’s a vertical asymptote at \(x=0\) and horizontal asymptotes at \(y=5\) and \(y=-5\).
• Determine the function’s domain by finding where the denominator equals zero. \(x = \pm1\) are excluded from the domain, so make sure to mark those points.
• Examine the behavior of the function as \(x\) approaches the asymptotes. For example, as \(x\) gets close to \(0\), \(f(x)\) drops towards \(-\infty\).
• Plot the intercepts; for \(f(x) = \frac{10x}{x^2-1}\), the only real intercept is at the origin, where \(x=0\) produces an \(f(x)\) value that is undefined, further hinting the presence of a vertical asymptote there.
• Draw the curve showing these behaviors graphically, bending towards the asymptotes, and illustrating the infinite surges and end behaviors approaching horizontal asymptotes.

Graph sketching combines mathematical analysis with visual representation, offering insight into the complex dynamics of functions and their limits.