The Natural Log Function as a Limit

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

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

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