Problem 6 - Calculus Early Transcendentals

Question

\(5-10\) Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions. $$ \begin{array}{l}{\lim _{x \rightarrow 0^{+}} f(x)=\infty, \quad \lim _{x \rightarrow 0^{-}} f(x)=-\infty, \quad \lim _{x \rightarrow \infty} f(x)=1} \\\ {\lim _{x \rightarrow-\infty} f(x)=1}\end{array} $$

Step-by-Step Solution

Answer

Use the function \(f(x) = \frac{2x}{x^2 + 1} + 1\) to meet the conditions.

Step 1: Understand the Limits

The limits tell us the behavior of the function as the input approaches specific values:- As \(x\) approaches \(0^+\), the function \(f(x)\) goes to infinity.- As \(x\) approaches \(0^-\), the function \(f(x)\) goes to negative infinity.- As \(x\) approaches infinity, the function \(f(x)\) approaches 1.- As \(x\) approaches negative infinity, the function \(f(x)\) also approaches 1.

Step 2: Define Function Characteristics

From the limit behavior, we see that the function is likely to have a vertical asymptote at \(x=0\). At large positive and negative \(x\), the function approaches a horizontal asymptote at \(y=1\). This suggests the function may take the form of a rational function.

Step 3: Choose a Function Form

A simple function that fulfills these conditions can be defined as: \[ f(x) = \frac{2x}{x^2 + 1} + 1 \]This form indicates the function will have a vertical asymptote at \(x=0\) and horizontal asymptotes at \(y=1\).

Step 4: Verify the Chosen Function

For \(x\rightarrow 0^+\), \(f(x) = \frac{2x}{x^2 + 1} + 1\) goes to infinity.For \(x\rightarrow 0^-\), \(f(x) = \frac{2x}{x^2 + 1} + 1\) goes to negative infinity.For \(x\rightarrow \pm\infty\), \(f(x)\) approaches \(1\).Thus, the selected function satisfies all conditions.

Step 5: Sketch the Graph

Based on the verified function \(f(x) = \frac{2x}{x^2 + 1} + 1\):1. Draw a vertical asymptote at \(x=0\).2. Sketch the curve approaching infinity as \(x\) approaches \(0^+\) and negative infinity as \(x\) approaches \(0^-\).3. As \(x\) approaches positive or negative infinity, draw the curve approaching the horizontal line \(y=1\).

Key Concepts

Vertical Asymptote

A vertical asymptote is a line that the graph of a function approaches but never actually touches. This usually happens when the function has a variable in the denominator that approaches zero, causing the value of the function to skyrocket to positive or negative infinity.

In the context of the exercise, as the function approaches zero from the right (\(x \to 0^+\)), the value of \(f(x)\) becomes infinite. Conversely, as \(x\) approaches zero from the left (\(x \to 0^-\)), \(f(x)\) plunges towards negative infinity.

This description indicates that there is a vertical asymptote at \(x = 0\). When graphing, this line is purely vertical at \(x = 0\), and you would draw the function going upwards and downwards towards this line, without crossing it.

Horizontal Asymptote

A horizontal asymptote occurs when a function approaches a particular value as \(x\) moves towards positive or negative infinity. Unlike vertical asymptotes, horizontal asymptotes can be crossed by the graph.

In our scenario, as \(x\) journeys towards both positive and negative infinity, \(f(x)\) approaches the value of 1. This implies a horizontal asymptote exists at \(y = 1\).

When sketching this on a graph, you would draw a line at \(y = 1\), and the curve of the function will tend to this line on both ends of the x-axis.

It's useful to remember that horizontal asymptotes are about end behavior - they tell us how the function behaves when \(x\) gets very large or very small, but not necessarily about what's happening for values in the middle range.

Function Behavior

To understand function behavior, consider how the function reacts to varying inputs. Generally, behavior can include considering where the function increases or decreases, and identifying any points where it could be undefined.

For the function in question, \(f(x) = \frac{2x}{x^2 + 1} + 1\), the critical information was derived from the limits provided:

• Approaches positive infinity as \(x\) goes to 0 from the positive side.
• Approaches negative infinity as \(x\) goes to 0 from the negative side.
• Tends towards 1 as \(x\) moves towards both infinity and negative infinity.

By examining these behaviors, you can predict the general shape of the function's graph. You understand that it crosses the y-axis at a certain point and then straddles the horizontal line \(y = 1\) at extreme values of \(x\).

Being aware of these characteristics helps solidify understanding and guide the drawing of the curve when plotting the graph.

Rational Functions

Rational functions are formed by dividing two polynomial functions where the numerator and the denominator are polynomials. They frequently have both vertical and horizontal asymptotes because of their division nature.

The function we analyzed, \(f(x) = \frac{2x}{x^2 + 1} + 1\), is a classic example of a rational function. Here:

• The numerator is \(2x\), a simple polynomial.
• The denominator is \(x^2 + 1\), another polynomial.

Rational functions like this often reveal interesting behaviors as they approach specific values on the x-axis, such as vertical asymptotes, and they often level off towards horizontal asymptotes at infinity.

Studying these elements within rational functions can provide deep insights into the dynamics of how these functions operate. Moreover, understanding their properties allows for accurate graphing and deeper comprehension of their long-term behavior.