# The Natural Log Function as a Limit

- Author:
- Albert Navetta

- Topic:
- Limits

Ever notice that the graphs of and are similar and wonder why? Adjust the sliders for , , and in the app below in small increments to see the effects on the graph of . The initial values are , , and , so After a few observations, check the

*set a=n*box so that , and move together and . Observe what happens to the graph of in relation to the graph of with each change.Recall that for any rational number , except , where
Here we are only interested in positive values of so
Next, since is continuous for we can assume that

.

(Note: this is assumed here, but not proven here.) The right side of the equation can be rewritten as The left side result is Because , choose the constant of integration, so that when Choosing , we have