Question

Ap Calculus question. Limit problem I think

Answer 1

\[\lim_{h \rightarrow 0}\frac{\ln(\frac{2+h}{2})}{h}=\lim_{h \rightarrow 0} \frac{\ln(\frac{2+h}{2})-\ln(\frac{2+0}{2}) }{h-0}=(\ln(\frac{2+h}{2}))'|_{h=0}\]

Answer 2

hi

Answer 3

So does h=0?

Answer 4

do you know the definition of derivative?
that is all I'm applying here

Answer 5

\[\lim_{x \rightarrow 0}\frac{f(x)-f(0)}{x-0}=(f(x))'|_{x=0}\]

Answer 6

you can use properties of log to rewrite what you have before differentiating

Answer 7

\[(\ln(2+h)-\ln(2))'|_{h=0}\]

Answer 8

so you only need to differentiate ln(2+h) w.r.t h and then plug in 0 for h when done

Answer 9

or a very similar other way to look at is like this:
\[\lim_{h \rightarrow 0}\frac{\ln(2+h)-\ln(2)}{h} \text{ we have 0/0 if we plug in 0 } \\ \text{ so use l'hosptial }\]

Answer 10

You need to do the derivative and then substitute h=0

Answer 11

Yes I know how to take the derivative it just seemed really complicated

Answer 12

it isn't that bad

Answer 13

derivative of ln(u) w.r.t x is u'/u

Answer 14

so derivative of ln(2+h) is (2+h)'/(2+h)

Answer 15

does that still seem complicated?

Answer 16

and then I plug in 0 for h right?

Answer 17

yes after doing the derivative there

Answer 18

Okay thank you! :)