Problem 79 - Algebra and Trigonometry

Question

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.

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: Analysis of the results

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