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anonymous

  • 5 years ago

lim (ln(2+x)-ln2)/(x)= 1/2 x--> 0

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  1. amistre64
    • 5 years ago
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    \[\frac{\ln(\frac{2+x}{2})}{x}\]

  2. amistre64
    • 5 years ago
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    \[\ln(1+x)/x = \ln(1+x)^{1/x}\]

  3. amistre64
    • 5 years ago
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    .... typoed it already

  4. anonymous
    • 5 years ago
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    no it is \[ \lim_{x \rightarrow 2} (\ln(2+X)- \ln2)\div x\]

  5. anonymous
    • 5 years ago
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    Just use l'hpital's rule.

  6. anonymous
    • 5 years ago
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    [ln (1 + x/2)] / x multiply and divide by 1/2 \[1/2[\ln (1 + x/2)]/x/2\]

  7. amistre64
    • 5 years ago
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    \[\ln(x+2)-\ln(x) = \ln(\frac{x+2}{x})\]

  8. anonymous
    • 5 years ago
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    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
    • 5 years ago
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    as x-> 2 we get ln(4/2)^(1/2) = ln(sqrt(2))

  10. anonymous
    • 5 years ago
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    amistre i think im right

  11. anonymous
    • 5 years ago
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    wtdu say anwar???

  12. amistre64
    • 5 years ago
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    him; youre right as far as x_.0 perhaps; but the question was amended

  13. anonymous
    • 5 years ago
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    yeah...

  14. anonymous
    • 5 years ago
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    Why don't use L'hopital's rule? It's going to be easy peasy.

  15. anonymous
    • 5 years ago
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    oh yeah but maybe he wants to show d actual method

  16. amistre64
    • 5 years ago
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    we both get to ln(1+x/2)^(1/x) :)

  17. anonymous
    • 5 years ago
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    no d qstn is ln (2+x) - ln2 whole upon x

  18. amistre64
    • 5 years ago
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    i know; and we both got to ln(1+ x/2)^(1/x)

  19. anonymous
    • 5 years ago
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    does ln 1/0 does not equal 1.

  20. amistre64
    • 5 years ago
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    as x-> 2 we get ln(sqrt(2))

  21. amistre64
    • 5 years ago
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    as x-> 0 we get ln(1)^(.000...0001)

  22. amistre64
    • 5 years ago
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    1^(tiny number) = 1 right?

  23. amistre64
    • 5 years ago
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    1 ^(any#) = 1 lol

  24. anonymous
    • 5 years ago
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    yes

  25. amistre64
    • 5 years ago
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    ln(1) = 0

  26. anonymous
    • 5 years ago
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    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
    • 5 years ago
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    The derivative as x goes to 0 is equal to 1/2

  28. anonymous
    • 5 years ago
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    I think we got the questioner confused :). @lovehap, Did you get what you asked about?

  29. anonymous
    • 5 years ago
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    yes it is \[\lim_{x \rightarrow 2}(\ln(2+x)-\ln(2))/ x\]

  30. anonymous
    • 5 years ago
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    \[f(x)=ln (x),f'(x)=\frac{1}{x}, f'(2)=\lim_{h\to0}\frac{ln(2+h)-ln(2)}{h}=\frac{1}{2}\]

  31. anonymous
    • 5 years ago
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    thank you satellite73!

  32. amistre64
    • 5 years ago
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    Dx(ln(x)) = 1/x f'(2) = 1/2.... yes

  33. anonymous
    • 5 years ago
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    welcome.

  34. anonymous
    • 5 years ago
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    ok thank you!

  35. KyanTheDoodle
    • one year ago
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    Past-satellite! You'll never believe what the future has in store for you!

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