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Chapter 4: 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
As \(x\) increases, the expression \(\left(1+\frac{1}{x}\right)^{x}\) seems to gradually approach a fixed number which is approximately 2.71828. This number is recognized in mathematics as the base of the natural logarithm, denoted by \(e\).

Step by step solution

01

Evaluation for x=10

Input \(x=10\) into the mathematical expression to receive \(\left(1+\frac{1}{10}\right)^{10}\). Calculate its value using a calculator.
02

Evaluation for x=100

Replace \(x\) again in the equation, now with the number \(100\), to obtain \(\left(1+\frac{1}{100}\right)^{100}\). Calculate the value of this expression as well.
03

Evaluation for other numbers

Repeat the same process for \(x=1000, 10000, 100000, 1000000\). Each time, insert the new value of \(x\) into the original mathematical expression and calculate its value.
04

Analysis of the results

Observe the outputs achieved in the previous steps and describe the general behavior of the expression as \(x\) increases.

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Most popular questions from this chapter

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