Problem 62 - Single Variable Calculus

Question

Let $$ f(x)=\frac{x^{2 n}-1}{x^{2 n}+1} $$ a. Plot the graphs of \(f\) for \(n=1,5,10,100\), and \(1000 .\) Do these graphs approach a "limiting" graph as \(n\) approaches infinity? b. Can you prove this result analytically?

Step-by-Step Solution

Answer

The graphs of \(f(x)=\frac{x^{2n}-1}{x^{2n}+1}\) for \(n=1,5,10,100\), and \(1000\) approach a limiting graph as \(n\) approaches infinity, which is given by: \(f(x) = \left\{ \begin{array}{ll} 1, & \mbox{if } x > 1 \\ 0, & \mbox{if } x = 1 \\ undefined, & \mbox{if } x = -1 \\ -1, & \mbox{if } x < -1 \end{array} \right.\) This result was proven analytically by taking the limit of the function as \(n\) approaches infinity and considering different cases for \(x\).

Step 1: Part a: Plotting graphs

To plot the graphs, we can substitute the given values of \(n\) into the function and plot each function in graphing software. Alternatively, we can use a website like Desmos or a graphing calculator to visualize the graphs. After doing this with the values \(n=1,5,10,100\), and \(1000\), we can observe that the graphs seem to be approaching a limiting graph as n increases. If we analyze the function further, we can see that when \(x=1\), \(f(x)=0\), and when \(x=-1\), \(f(x)=\frac{-2}{0}\), which is undefined. Additionally, as \(x\) approaches infinity, the function approaches 1, and as \(x\) approaches negative infinity, the function approaches -1. From these observations, it appears that the limiting graph of \(f(x)\) is: \(f(x) = \left\{ \begin{array}{ll} 1, & \mbox{if } x > 1 \\ 0, & \mbox{if } x = 1 \\ undefined, & \mbox{if } x = -1 \\ -1, & \mbox{if } x < -1 \end{array} \right.\)

Step 2: Part b: Analytical proof

Now, let's try to prove this result analytically. We want to find the limit of \(f(x)\) as \(n\) approaches infinity. To do that, let's rewrite the function as: \(f(x) =\frac{x^{2n}-1}{x^{2n}+1} = \frac{(x^{2n}-1)+(x^{2n}+1)-2x^{2n}}{x^{2n}+1}=\frac{2 - 2x^{2n}}{x^{2n} + 1}\) Now, let's find the limit of this function: \(\lim_{n\to\infty}\frac{2-2x^{2n}}{x^{2n}+1}\) We can split this limit into cases: 1. When \(x > 1\), \(x^{2n} \to\infty\) as \(n\to\infty\). Thus, the limit is \(\lim_{n\to\infty}\frac{2 - 2\cdot\infty}{\infty+1} = 1\). 2. When \(x = 1\), the function is always 0, so the limit is 0. 3. When \(-11\), so \(x^{2n}\to\infty\) as \(n\to\infty\). Thus, the limit is \(\lim_{n\to\infty}\frac{2-2\cdot-\infty}{-\infty+1}=-1\). 5. When \(x=-1\), the function is undefined. From these cases, we can conclude that the limiting graph as \(n\) approaches infinity is the same as the one we found analytically: \(f(x) = \left\{ \begin{array}{ll} 1, & \mbox{if } x > 1 \\ 0, & \mbox{if } x = 1 \\ undefined, & \mbox{if } x = -1 \\ -1, & \mbox{if } x < -1 \end{array} \right.\)

Key Concepts

Limits

Limits are fundamental in understanding how a function behaves as it approaches a certain point, especially when precise calculation at that point is difficult or impossible.
The limit of a function is the value that the function's output gets closer to as the input approaches a specific point.
In our case, we are examining the function \( f(x) = \frac{x^{2n}-1}{x^{2n}+1} \) to see how it behaves as \( n \) gets very large.
This means we need to analyze how the expression's value changes when \( n \to \infty \), at various values of \( x \).
Here are some key aspects to consider when evaluating limits:

• As \( x > 1 \), the power \( x^{2n} \) becomes very large, leading \( f(x) \) towards 1.
• When \( -1 < x < 1 \), \( x^{2n} \) will tend to 0, causing \( f(x) \) also to approach 1.
• Special points like \( x = 1 \) have \( f(x) = 0 \) directly and \( x = -1 \) causes \( f(x) \) to be undefined because of division by zero.

Understanding these principles allows us to determine the behavior of \( f(x) \) as \( n \) increases, leading to the limiting graph.

Graphing

Graphing is an essential tool in calculus to visually represent the behavior of functions.
For the function \( f(x) = \frac{x^{2n}-1}{x^{2n}+1} \), plotting its graph for increasing values of \( n \) helps us visualize and understand the trend or limit that the graphs approach.
To plot these graphs:

• Choose different large values of \( n \) like 1, 5, 10, and even bigger ones like 100, and 1000.
• Use graphing software such as Desmos for a clear picture.

As \( n \) increases,

• The graphs will flatten, showing how they approach a particular shape or line.
• The function becomes almost constant for values between -1 and 1, whereas it sharply transitions outside this region.

This visualization helps confirm that the plotted graphs are indeed moving towards the "limiting" graph as \( n \) increases.

Infinity

Infinity is a concept within mathematics that refers to a quantity without bound or end.
When analyzing the function \( f(x) \), the notion of approaching infinity is crucial to understanding how the function behaves when \( n \to \infty \) or when \( x \) approaches very large positive or negative values.
For example:

• As \( n \to \infty \), the term \( x^{2n} \) takes on extremely large values when \( |x| > 1 \), and approaches zero when \( |x| < 1 \).
• This characteristic creates a stark contrast in the behavior of the graph, where it appears steady away from discontinuities at \( x = 1 \) and undefined at \( x = -1 \).

Infinity, in this context, allows us to consider limits and the ultimate behavior of functions without dealing with finite boundaries.

Function Behavior

Understanding function behavior involves examining how a function changes across its domain, especially as parameters or variables approach specific values.
For the given function \( f(x) = \frac{x^{2n}-1}{x^{2n}+1} \), analyzing its behavior as \( n \to \infty \) provides insight into its long-term trends.
Critical points and cases are:

• \( x > 1 \), where the function tends towards 1 as these values dominate the expression.
• For \( -1 < x < 1 \), it also approaches 1, indicating a constant pattern in large \( n \) limits.
• At \( x = 1 \) and \( x = -1 \), unique behaviors like holding at 0 or becoming undefined, highlight transition areas in the function.

This exploration helps one predict where the function stays stable or changes sharply, forming a base for understanding deeper properties in calculus.