The Natural Log Function as a Limit

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

. (1)

(Note: this assumption works here, but is not necessarily true in general.) The right side of equation (1) can be rewritten as . Substituting for the right side of equation (1) and combining constants,

Because , choose the constant of integration, so that when Choosing ,