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

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(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