Problem 7 - Calculus Early Transcendentals

Question

Analyzing infinite limits numerically Compute the values of \(f(x)=\frac{x+1}{(x-1)^{2}}\) in the following table and use them to determine \(\lim _{x \rightarrow 1} f(x)\) $$\begin{array}{|c|c|c|c|} \hline x & \frac{x+1}{(x-1)^{2}} & x & \frac{x+1}{(x-1)^{2}} \\ \hline 1.1 & & 0.9 & \\ \hline 1.01 & & 0.99 & \\ \hline 1.001 & & 0.999 & \\ \hline 1.0001 & & 0.9999 & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Answer: The limit of the function as \(x\) approaches \(1\) is infinity, \(\lim _{x \rightarrow 1} f(x) = \infty\).

Step 1: Compute \(f(x)\) values in the table for given \(x\) values.

We will compute the values of \(f(x)\) using the given formula for each value of \(x\).

Step 2: Analyze the table values to understand the behavior of \(f(x)\) as \(x\) approaches \(1\).

We will inspect the computed values of \(f(x)\) in the table and check whether there is a pattern that could help us to determine the limit of the function as \(x\) approaches \(1\).

Step 3: Determine the limit.

Based on the analysis in Step 2, we will make a conclusion regarding the limit \(\lim _{x \rightarrow 1} f(x)\). Now we are ready to find the limit. Let's start with computing the table values.

Step 1: Compute \(f(x)\) values in the table for given \(x\) values.

We need to compute \(f(x) = \frac{x+1}{(x-1)^2}\) for the given \(x\) values: For \(x=1.1\), we get: \(f(1.1)=\frac{1.1+1}{(1.1-1)^{2}}=\frac{2.1}{0.01^2}=21000\) For \(x=1.01\), we get: \(f(1.01)=\frac{1.01+1}{(1.01-1)^{2}}=\frac{2.01}{0.01^2}=201000\) For \(x=1.001\), we get: \(f(1.001)=\frac{1.001+1}{(1.001-1)^{2}}=\frac{2.001}{0.001^2}=2001000\) For \(x=1.0001\), we get: \(f(1.0001)=\frac{1.0001+1}{(1.0001-1)^{2}}=\frac{2.0001}{0.0001^2}=200010000\) For \(x=0.9\), we get: \(f(0.9)=\frac{0.9+1}{(0.9-1)^{2}}=\frac{1.9}{0.1^2}=190\) For \(x=0.99\), we get: \(f(0.99)=\frac{0.99+1}{(0.99-1)^{2}}=\frac{1.99}{0.01^2}=19900\) For \(x=0.999\), we get: \(f(0.999)=\frac{0.999+1}{(0.999-1)^{2}}=\frac{1.999}{0.001^2}=1999000\) For \(x=0.9999\), we get: \(f(0.9999)=\frac{0.9999+1}{(0.9999-1)^{2}}=\frac{1.9999}{0.0001^2}=199990000\) Here is the completed table with the calculated values: $$\begin{array}{|c|c|c|c|} \hline x & \frac{x+1}{(x-1)^{2}} & x & \frac{x+1}{(x-1)^{2}} \\\ \hline 1.1 & 21000 & 0.9 & 190 \\\ \hline 1.01 & 201000 & 0.99 & 19900 \\\ \hline 1.001 & 2001000 & 0.999 & 1999000 \\\ \hline 1.0001 & 200010000 & 0.9999 & 199990000 \\\ \hline \end{array}$$

Step 2: Analyze the table values to understand the behavior of \(f(x)\) as \(x\) approaches \(1\).

By examining the table, we can see that as \(x\) approaches \(1\), the values of \(f(x)\) are increasing towards infinity without limitations for both the left and right sides. The sequence of \(f(x)\) values increases very fast as \(x\) moves closer to \(1\).

Step 3: Determine the limit.

Based on the analysis in Step 2, we can conclude that the limit of the function as \(x\) approaches \(1\) is infinity: $$\lim _{x \rightarrow 1} f(x) = \infty$$

Key Concepts

Function Behavior

When analyzing the behavior of functions, it's essential to understand how they act as the input (or independent variable, x) changes. The function in our example is given by \[f(x) = \frac{x+1}{(x-1)^2}\] As x gets closer and closer to 1, this function behaves in a particular way that causes its output to change dramatically. Function behavior can give us insights into the nature of the graph and its tendencies:

• Rising steeply or declining swiftly
• Leveling out
• Approaching certain points (called limits)

In this exercise, as x approaches 1, both from the left (values like 0.9999) and the right (values like 1.0001), we notice the function's value increases without bound. This behavior implies that the function is tending towards infinity, highlighting the significant role of the denominator becoming remarkably small near the value of 1.

Numerical Analysis

Numerical analysis involves the use of numerical approximations to understand the behavior of functions in areas where analytical solutions might be challenging or impossible to obtain. In this problem, we are tasked with computing function values for specific x-values close to 1: - For x-values above 1 (like 1.1, 1.01), the function produces very large positive numbers. - For x-values below 1 (like 0.9, 0.99), the function also results in large positive numbers, although generally less than those above 1. By carefully choosing these x-values, we can see the function's trend as x approaches 1 from either side. This method shows how the function's outputs are not just increasing but doing so at an accelerating pace. Numerical analysis allows us to approximate behavior over small increments, revealing intricate function details without needing an explicit formula-based proof every time.

Limit Evaluation

Limit evaluation is a fundamental concept in calculus that deals with determining how a function behaves as it approaches a certain point. In the given problem, we need to find \[\lim_{x \to 1} f(x)\]To evaluate this limit, we calculate \[f(x) = \frac{x+1}{(x-1)^2}\]for various values of x, ever closer to 1, from both directions. By evaluating these limits numerically, we see that they consistently head towards infinity:- As x approaches 1 from the right (e.g., 1.001, 1.0001), the values explode upwards to larger numbers quickly.- Similarly, from the left (0.9999, 0.9998), they also trend upwards, confirming the behavior is consistent from both sides.Evaluating the limit helps conclude that as x approaches 1, the function's output is indeed moving towards infinity, thereby demonstrating the function's undefined nature at that exact point.

Approaching Values

"Approaching values" refers to understanding what happens to a function as we get near a specific input value. In the infinite limit problem, approaching the value 1 is crucial to uncovering the behavior of the function. The key elements in understanding approaching values are:

• From the right side (values like 1.01, 1.001), the output grows rapidly.
• From the left side (values like 0.99, 0.999), the output also grows, albeit sometimes slightly less aggressively.
• The closer x gets to 1, the smaller (x-1) becomes, causing the denominator to shrink, and thus the entire fraction to increase significantly.

Approaching values lets us see that no matter how we try to visualize the function at x = 1, it always tends unboundedly upwards, reinforcing the conclusion that the limit does not settle into a fixed number but instead shoots to infinity. It's a demonstration of how approaching values, incremental by nature, tease out infinite or undefined behaviors in functions.