How it works ?
Consider the function f(x)=(1+(1/x))x .
What happens as the input (x) gets bigger and bigger? The exponent will get infinitely large, but the base, 1+(1/x), will approach the value 1 because as x gets bigger (i.e. “approaches infinity”), the ratio1/x approaches zero.
Thus, as x approaches infinity, we have a limit that structurally looks like 1^(“infinity”).
What do you think will “WIN” here, so to speak?
Will the “BIG-NESS” of the exponent cause the outputs of this function to skyrocket (approach positive infinity) OR will the “SMALLNESS of THE BASE — that approaches a limiting value of 1) “win” and cause this function to have a finite “maximum value” that gets approached?
Interact with the applet for a few minutes. Then answer the questions that follow.