# 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?