Chapter 2 - Limits - 2.2 Definitions of Limits - 2.2 Exercises - Page 66: 14

a) $x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f(x)$ $0.01\ \ \ \ \ \ \ \ \ 2.70481$ $0.001\ \ \ \ \ \ \ 2.71692$ $0.0001\ \ \ \ \ 2.71815$ $0.00001\ \ \ 2.71827$ $x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f(x)$ $-0.01\ \ \ \ \ \ \ \ 2.73200$ $-0.001\ \ \ \ \ \ 2.71964$ $-0.0001\ \ \ \ 2.71842$ $-0.00001\ \ 2.71829$ b) As x approaches 0, f(x) appears to approach 2.71818. c) This limit appears to approach the mathematical constant known as Euler's number.

a) Use a calculator to evaluate the function to each of the given values. $f(x)=(1+x)^{\frac{1}{x}}$ b) Reviewing the table of results you can see that the solutions, as they approach 0 from both the positive and negative direction, seem to be converging on a value of 2.71828. c) 2.7182818284590452353602874713527 is known as Euler's number.