One Special Limit
Consider the function .
What happens as the input (x) gets bigger and bigger? The exponent will get infinitely large, but the base, , will approach the value 1 because as x gets bigger (i.e. "approaches infinity"), the ratio approaches zero.
Thus, as x approaches infinity, we have a limit that structurally looks like 1^("infinity").
So.....
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.
1.
After dragging the slider all the way to the right, drag the purple point as far to the left as you can. BE SURE TO PAN & ZOOM as you do! Is there a value that the function seems to approach as the input (x) gets smaller and smaller?
2.
After dragging the slider all the way to the right, drag the brown point as far to the right as you can. Be sure to PAN & ZOOM as you do! Is there a value that the function seems to approach as the input (x) gets larger and larger?
3.
If your answers to (1) & (2) were both "yes", how do these values compare with each other? What is each approximate value?