anonymous
  • anonymous
lim (ln(2+x)-ln2)/(x)= 1/2 x--> 0
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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amistre64
  • amistre64
\[\frac{\ln(\frac{2+x}{2})}{x}\]
amistre64
  • amistre64
\[\ln(1+x)/x = \ln(1+x)^{1/x}\]
amistre64
  • amistre64
.... typoed it already

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anonymous
  • anonymous
no it is \[ \lim_{x \rightarrow 2} (\ln(2+X)- \ln2)\div x\]
anonymous
  • anonymous
Just use l'hpital's rule.
anonymous
  • anonymous
[ln (1 + x/2)] / x multiply and divide by 1/2 \[1/2[\ln (1 + x/2)]/x/2\]
amistre64
  • amistre64
\[\ln(x+2)-\ln(x) = \ln(\frac{x+2}{x})\]
anonymous
  • 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
amistre64
  • amistre64
as x-> 2 we get ln(4/2)^(1/2) = ln(sqrt(2))
anonymous
  • anonymous
amistre i think im right
anonymous
  • anonymous
wtdu say anwar???
amistre64
  • amistre64
him; youre right as far as x_.0 perhaps; but the question was amended
anonymous
  • anonymous
yeah...
anonymous
  • anonymous
Why don't use L'hopital's rule? It's going to be easy peasy.
anonymous
  • anonymous
oh yeah but maybe he wants to show d actual method
amistre64
  • amistre64
we both get to ln(1+x/2)^(1/x) :)
anonymous
  • anonymous
no d qstn is ln (2+x) - ln2 whole upon x
amistre64
  • amistre64
i know; and we both got to ln(1+ x/2)^(1/x)
anonymous
  • anonymous
does ln 1/0 does not equal 1.
amistre64
  • amistre64
as x-> 2 we get ln(sqrt(2))
amistre64
  • amistre64
as x-> 0 we get ln(1)^(.000...0001)
amistre64
  • amistre64
1^(tiny number) = 1 right?
amistre64
  • amistre64
1 ^(any#) = 1 lol
anonymous
  • anonymous
yes
amistre64
  • amistre64
ln(1) = 0
anonymous
  • 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
anonymous
  • anonymous
The derivative as x goes to 0 is equal to 1/2
anonymous
  • anonymous
I think we got the questioner confused :). @lovehap, Did you get what you asked about?
anonymous
  • anonymous
yes it is \[\lim_{x \rightarrow 2}(\ln(2+x)-\ln(2))/ x\]
anonymous
  • anonymous
\[f(x)=ln (x),f'(x)=\frac{1}{x}, f'(2)=\lim_{h\to0}\frac{ln(2+h)-ln(2)}{h}=\frac{1}{2}\]
anonymous
  • anonymous
thank you satellite73!
amistre64
  • amistre64
Dx(ln(x)) = 1/x f'(2) = 1/2.... yes
anonymous
  • anonymous
welcome.
anonymous
  • anonymous
ok thank you!
KyanTheDoodle
  • KyanTheDoodle
Past-satellite! You'll never believe what the future has in store for you!

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