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Chapter 3: Problem 79

Use a calculator to evaluate \(\left(1+\frac{1}{x}\right)^{x}\) for \(x=10,100,1000\) \(10,000,100,000,\) and \(1,000,000 .\) Describe what happens to the expression as \(x\) increases.

Short Answer

Expert verified
After performing each calculation, you'll notice that as \(x\) increases, the expression \(\left(1+\frac{1}{x}\right)^{x}\) approaches a specific value, which is approximately \(2.71828\), also known as the number \(e\). And once you'd hit higher values like \(100,000\) or \(1,000,000\), the result should stay quite constant at that value \(2.71828\) more or less.

Step by step solution

01

Set up the expression

First, write down the expression \(\left(1+\frac{1}{x}\right)^{x}\). This will be our starting point for the calculations.
02

Substitute each value of \(x\)

Starting with \(x = 10\), substitute \(10\) into the equation and use a calculator to solve \(\left(1+\frac{1}{10}\right)^{10}\). Then, do the same for each of the remaining values of \(x = 100,1000,10000,100000\) and \(1000000\).
03

Observe the trend

Take note of the results obtained in step 2. By comparing each result, you can observe the trend or pattern that the expression is following as \(x\) increases.

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