Consider the function [math]f\left(x\right)=\left(1+\frac{1}{x}\right)^x[/math].[br][br]What happens as the input ([i]x[/i]) gets bigger and bigger? The exponent will get infinitely large, but the base, [math]1+\frac{1}{x}[/math], will approach the value 1 because as [i]x[/i] gets bigger (i.e. "approaches infinity"), the ratio [math]\frac{1}{x}[/math] approaches zero. [br][br]Thus, as [i]x[/i] approaches infinity, we have a limit that structurally looks like 1^("infinity"). [br][br]So.....[br][br][color=#0000ff][b]What do you think will "WIN" here, so to speak? [br][br]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? [/b][/color][br][br][color=#38761d][b]Interact with the applet for a few minutes. [br]Then answer the questions that follow. [/b][/color]
After dragging the slider all the way to the right, [color=#9900ff][b]drag the purple point as far to the left as you can[/b][/color]. BE SURE TO PAN & ZOOM as you do! Is there a value that the function seems to approach [color=#9900ff][b]as the input ([i]x[/i]) gets smaller and smaller? [/b][/color]
After dragging the slider all the way to the right, [b][color=#980000]drag the brown point as far to the right as you can[/color][/b]. Be sure to PAN & ZOOM as you do! Is there a value that the function seems to approach [color=#980000][b]as the input ([i]x[/i]) gets larger and larger? [/b][/color]
If your answers to (1) & (2) were both [b]"yes"[/b], how do these values compare with each other? What is each approximate value?