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Chapter 2: Problem 45

Find the limit. Use the TABLE feature and start with \(\Delta \mathrm{Tbl}=0.1\). Then move to \(0.01\), \(0.001\), and \(0.0001\). When you think you know the limit, graph and use the TRACE feature to verify your assertion. Then try to verify algebraically. $$\lim _{x \rightarrow 1} \frac{x-\sqrt[4]{x}}{x-1}$$

Short Answer

Expert verified
The limit is \frac{4}{4}. =1

Step by step solution

01

- Understand the Function

The given function is \ \(f(x)=\frac{x-\sqrt[4]{x}}{x-1}\). We need to find the limit as \(x\) approaches 1.
02

- Create a Table with \(\Delta Tbl = 0.1\)

Create a table to evaluate \(f(x)\) for small increments. Start with \(0.1\):\x = 0.9, 0.99, 0.999, 0.9999, 1.1, 1.01, 1.001, 1.0001\Compute the values: f(0.9), f(0.99), f(0.999), f(0.9999), f(1.1), f(1.01), f(1.001), f(1.0001).
03

- Evaluate the Limit with \(\Delta Tbl = 0.01\)

Repeat the process with steps of \(0.01\):\x = 0.91, 0.99, 0.999, 0.9999, 1.1, 1.01, 1.001, 1.0001\Compute the values: f(0.91), f(0.99), f(0.999), f(0.9999), f(1.1), f(1.01), f(1.001), f(1.0001).
04

- Evaluate the Limit with \(\Delta Tbl = 0.001\)

Repeat the process with steps of \(0.001\):\x = 0.999, 0.9999, 1.001, 1.0001\Compute the values: f(0.999), f(0.9999), f(1.001), f(1.0001).
05

- Evaluate the Limit with \(\Delta Tbl = 0.0001\)

Repeat the process with steps of \(0.0001\):\x = 0.9999, 1.0001\Compute the values: f(0.9999), f(1.0001).
06

- Observe the Trend from the Table

Notice the values of \(f(x)\) as \(x\) gets closer and closer to 1. The values should cluster around a specific number.
07

- Verify with the Graph and TRACE Feature

Graph \(f(x)=\frac{x-\sqrt[4]{x}}{x-1}\) on a graphing calculator. Use the TRACE feature to visually verify the clustering point as x approaches 1.
08

- Verify Algebraically

Rewrite the function \( \ \frac{x-\sqrt[4]{x}}{x-1} \) and use algebraic techniques to simplify. Factor the numerator \[ x - \sqrt[4]{x} = (\sqrt[4]{x})^4 - (\sqrt[4]{x}) = (\sqrt[4]{x}-1)(\sqrt[4]{x}+1)(\sqrt[4]{x}^2+1)\]. Simplify the expression to find the limit.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit Evaluation
Evaluating the limit of a function as it approaches a certain value is a fundamental concept in calculus. For the given function \(f(x)=\frac{x-\sqrt[4]{x}}{x-1}\), we need to find its limit as \(x\) approaches 1. This involves understanding how the function behaves very close to this point. By examining values of the function at points increasingly near \(x=1\), such as using a table of values, we can observe whether they cluster around a specific value. In this case, the limit is seen as \(x\) approaches 1 from both the left and right sides. Inputting smaller increments closer to 1 helps identify a trend in the output values, leading to the deduced limit.
Algebraic Simplification
Algebraic simplification is a powerful technique used to evaluate limits. For the function \(f(x)=\frac{x-\sqrt[4]{x}}{x-1}\), directly substituting \(x=1\) causes an indeterminate form \(\frac{0}{0}\). To resolve this, we rework the expression algebraically. By factoring the numerator as \( x - \sqrt[4]{x} = (\sqrt[4]{x})^4 - \sqrt[4]{x} \), we get \( (\sqrt[4]{x}-1)(\sqrt[4]{x}+1)(\sqrt[4]{x}^2+1)\). This way, the expression significantly simplifies and helps in determining the limit. After the simplification, re-evaluating at \(x=1\) gives the actual limit without indeterminate forms.
Graphing Calculator Techniques
Using graphing calculators can greatly aid in understanding the behavior of complex functions. For \(f(x)=\frac{x-\sqrt[4]{x}}{x-1}\), you can input the function into the graphing calculator and utilize the TRACE feature. This feature allows you to visually follow the function's output as \(x\) approaches 1. By closely observing the plotted graph near this value, the function's trend and behavior become clear, visually confirming the limit derived via numerical evaluation or algebraic simplification. Graphing provides a quick and intuitive validation of the limit without manual computation of values.
Numerical Methods
Numerical methods are practical for approximating the limits of functions. By creating tables with small steps of \(0.1\), \(0.01\), \(0.001\), and finally \(0.0001\), we compute the function's values at these steps close to \(x=1\). For \(f(x)=\frac{x-\sqrt[4]{x}}{x-1}\), creating a table of \(f(x)\) at points like \(0.9\), \(0.99\), \(0.999\), \(0.9999\), \ and \(1.1\), allows us to observe the outcomes as \(x\) gets closer to 1. Numerical precision increases with smaller increments, making the results more reliable. Observing these values shows a trend around a specific number, confirming what the graphing and algebra show about the limit.

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