Problem 11 - Precalculus with Limits

Question

In Exercises 9-28, find the limit (if it exists). If the limit does not exist, explain why. Use a graphing utility to verify your result graphically. \\[\lim_{x\to \infty} \left(\dfrac{1-x}{1+x} \right) \\]

Step-by-Step Solution

Answer

The limit of the function as \(x\) goes to infinity is -1.

Step 1: Divide the top and bottom by \(x\)

To solve this problem, first divide the top and the bottom by \(x\). It will result in \(f(x)=\dfrac{1/x - 1}{1/x + 1}\). Now, as \(x\) goes to infinity, \(1/x\) goes to 0, and this simplifies finding the limit.

Step 2: Calculate the limit

Next, substitute the values of \(1/x\) with 0 in the equation \(f(x) = \dfrac{0-1}{0+1}\). Therefore, the limit of the function \(f(x)\) as \(x\) goes to infinity is -1.

Step 3: Verify graphically

Now, plot the function \(f(x) = \dfrac{1-x}{1+x}\) using a graphing utility. As \(x\) approaches infinity, observe how the y-value approaches -1, which is consistent with our previous calculations.

Key Concepts

Infinite Limit

When we talk about infinite limits in calculus, we're interested in what happens to a function's values as the input grows very large or very small. In our exercise, the limit to find is \( \lim_{x \to \infty} \left(\frac{1-x}{1+x}\right) \). We consider the behavior as \( x \) becomes extremely large—this is what 'infinite' refers to.
To handle an infinite limit, often, we simplify the expression by factoring out variables or using algebraic manipulation. For this function, we divide the numerator and the denominator by \( x \). This gives us a simplified version of the function that makes it easier to see how it behaves when \( x \) grows. This leads to \( f(x) = \frac{1/x - 1}{1/x + 1} \). As \( x \to \infty \), the terms \( 1/x \) approach 0, simplifying the solution and making it clear that the function approaches -1.
The infinite limit is a valuable tool. It helps predict the behavior of equations at their extremes. Recognizing these patterns makes it easier to predict how different functions will act without graphing them or plugging in large numbers.

Asymptotic Behavior

The concept of asymptotic behavior is closely related to infinite limits. An asymptote is a line that a graph approaches but never actually reaches. This is important in understanding the long-term behavior of functions.
In our exercise, as \( x \) goes to infinity, the function \( f(x) = \frac{1-x}{1+x} \) approaches the line \( y = -1 \). This means that the graph of the function has a horizontal asymptote at \( y = -1 \). Simply put, as you go further out on the x-axis, the graph of the function gets very close to the line \( y = -1 \), but it doesn’t actually touch or cross it.
Recognizing asymptotic behavior helps us understand the trends in a graph without needing exact data points. It provides insight into the limits of a function, offering a simpler way to predict how the function behaves at extreme values.

Graphical Verification

Graphical verification is a step in the process of checking one's mathematical work. It involves using a graph to confirm analytical results. For infinite limits and asymptotic behavior, plotting a graph gives a visual representation of what the math describes.
In our example, using a graphing utility to plot the function \( f(x) = \frac{1-x}{1+x} \), we can observe how the y-values of the graph behave as \( x \) becomes larger. As expected, the graph trends toward \( -1 \), confirming the analytical solution of the limit \( \lim_{x \to \infty} \left(\frac{1-x}{1+x}\right) = -1 \).
The graphical approach can clarify how functions behave, showing limiting behavior like asymptotes in an intuitive manner. Even when our algebraic techniques are solid, a graph can reinforce understanding by vividly displaying what happens as \( x \) increases or decreases greatly.