Question

Find the lim [(ln(x+5) - ln5) / x ] as x approaches 0

Answer 1

looks like a l'hopital problem

Answer 2

ok so there is an interesting rule that applies called L'hopital's rule. Since both top and bottom =0 if you plug in 0 for x, you can take the derivative of top and bottom until something doesn't equal zero.
In this case, you would get
\[d(\ln(x+5)-\ln(5)/x)=d(\ln(x+5)-\ln(5))/d(x)\]
which is
\[1/(x+5)/1|x=0\]
which is
\[1/5\]

Answer 3

well actually not really. this is also the derivative of log at x = 5

Answer 4

and since
\[\frac{d}{dx}\ln(x)=\frac{1}{x}\] you get
\[\frac{1}{5}\]

Answer 5

I actually haven't learned how to find derivatives yet. Is there a way to do it without that?

Answer 6

l'hopital works too of course. if you cannot use derivatives, then you can only guess

Answer 7

no not really. you will have to plug in numbers close to 0 and see what you get

Answer 8

take
\[\frac{\ln(5.1)=\ln(5)}{.1}\] for example

Answer 9

i meant
\[\frac{\ln(5.1)-\ln(5)}{.1}\]

Answer 10

without using derivatives you can neither use l'hopital not recognize this as a derivative
you cannot do it with algebra

Answer 11

*nor

Answer 12

ok, gotcha, thanks(: I'l just do the plug in method.

Answer 13

yup and see that you get numbers close and closer to 0.2