Answer
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.
Work Step by Step
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.
