## anonymous 5 years ago lim (ln(2+x)-ln2)/(x)= 1/2 x--> 0

1. amistre64

$\frac{\ln(\frac{2+x}{2})}{x}$

2. amistre64

$\ln(1+x)/x = \ln(1+x)^{1/x}$

3. amistre64

4. anonymous

no it is $\lim_{x \rightarrow 2} (\ln(2+X)- \ln2)\div x$

5. anonymous

Just use l'hpital's rule.

6. anonymous

[ln (1 + x/2)] / x multiply and divide by 1/2 $1/2[\ln (1 + x/2)]/x/2$

7. amistre64

$\ln(x+2)-\ln(x) = \ln(\frac{x+2}{x})$

8. anonymous

now putting x-> 0 it becomes a standard integral of form (ln 1)/0 which is equal to 1, so the ans is 1/2 x 1 = 1/2

9. amistre64

as x-> 2 we get ln(4/2)^(1/2) = ln(sqrt(2))

10. anonymous

amistre i think im right

11. anonymous

wtdu say anwar???

12. amistre64

him; youre right as far as x_.0 perhaps; but the question was amended

13. anonymous

yeah...

14. anonymous

Why don't use L'hopital's rule? It's going to be easy peasy.

15. anonymous

oh yeah but maybe he wants to show d actual method

16. amistre64

we both get to ln(1+x/2)^(1/x) :)

17. anonymous

no d qstn is ln (2+x) - ln2 whole upon x

18. amistre64

i know; and we both got to ln(1+ x/2)^(1/x)

19. anonymous

does ln 1/0 does not equal 1.

20. amistre64

as x-> 2 we get ln(sqrt(2))

21. amistre64

as x-> 0 we get ln(1)^(.000...0001)

22. amistre64

1^(tiny number) = 1 right?

23. amistre64

1 ^(any#) = 1 lol

24. anonymous

yes

25. amistre64

ln(1) = 0

26. anonymous

This is the definition of the derivative of ln(x) at x = 2. Since the derivative of ln(x) = 1/x, at 2 you get 1/2

27. anonymous

The derivative as x goes to 0 is equal to 1/2

28. anonymous

I think we got the questioner confused :). @lovehap, Did you get what you asked about?

29. anonymous

yes it is $\lim_{x \rightarrow 2}(\ln(2+x)-\ln(2))/ x$

30. anonymous

$f(x)=ln (x),f'(x)=\frac{1}{x}, f'(2)=\lim_{h\to0}\frac{ln(2+h)-ln(2)}{h}=\frac{1}{2}$

31. anonymous

thank you satellite73!

32. amistre64

Dx(ln(x)) = 1/x f'(2) = 1/2.... yes

33. anonymous

welcome.

34. anonymous

ok thank you!

35. KyanTheDoodle

Past-satellite! You'll never believe what the future has in store for you!