Question

I am having a hard time continuing to the next step with this lim h -->0 f(x+h)-f(x)/h when f(X) = ln x

Answer 1

i know it will be \[\lim_{h \rightarrow 0} (ln (x+h) - lnx) / h\]

Answer 2

but i do not know the next step after that.

Answer 3

very difficult i think the next step is use the property of logariths to make ln( (x+h)/x )

Answer 4

Use a logarithm property. In the numerator, you have
\[\ln(x+h)-\ln x=\ln\frac{x+h}{x}=\ln\left(1+\frac{h}{x}\right)\]
\[\lim_{h\to0}\frac{\ln\left(1+\frac{h}{x}\right)}{h}\]
Let \(n=\dfrac{1}{h}\). As \(h\to0\), we have \(n\to\infty\).
\[\lim_{n\to\infty}n\ln\left(1+\frac{1}{nx}\right)\\
\lim_{n\to\infty}\ln\left(1+\frac{1}{nx}\right)^n\]
Next, you multiply by \(\dfrac{x}{x}\):
\[\lim_{n\to\infty}\frac{x}{x}\ln\left(1+\frac{1}{nx}\right)^n\\
\lim_{n\to\infty}\frac{1}{x}\ln\left(1+\frac{1}{nx}\right)^{nx}\\
\frac{1}{x}\lim_{n\to\infty}\ln\left(1+\frac{1}{nx}\right)^{nx}\]
Since the natural logarithm function is continuous for \(n>0\), you can "pull out" the log:
\[\frac{1}{x}\ln\left[\lim_{n\to\infty}\left(1+\frac{1}{nx}\right)^{nx}\right]\]

Answer 5

that demostration has a lot of math tricks well here is a video about that problem
http://www.youtube.com/watch?v=ufyWicnng_E

Answer 6

Thank You so much!