# One Special Limit

- Author:
- Tim Brzezinski

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?