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anonymous

  • one year ago

lim t->0 [1/(t√(1+t))-(1/t)]

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  1. anonymous
    • one year ago
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    \[\lim_{x \rightarrow 16} (\frac{ 1 }{ t \sqrt{1+t} } - \frac{ 1 }{ t })\]

  2. dinamix
    • one year ago
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    (1-(17)^1/2 /(17)^1/2) 1/16

  3. Michele_Laino
    • one year ago
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    hint: your function, is continue at t=16

  4. anonymous
    • one year ago
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    *as t approaches 0... sorry

  5. Michele_Laino
    • one year ago
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    you have to replace t with 16 only

  6. dinamix
    • one year ago
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    its \[-\]

  7. dinamix
    • one year ago
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    when t---->0

  8. anonymous
    • one year ago
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    the answer is -1/2

  9. anonymous
    • one year ago
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    yes, the answer is -1/2 @vvvbb

  10. anonymous
    • one year ago
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    yes... so how do I get there?

  11. anonymous
    • one year ago
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    |dw:1440949656458:dw| choose common denominator and simplify

  12. anonymous
    • one year ago
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    then you get\[\frac{ 1-\sqrt{1+t} }{ t \sqrt{1+t}}\]

  13. anonymous
    • one year ago
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    |dw:1440949891651:dw|

  14. idku
    • one year ago
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    \[\lim_{t \rightarrow 0}\left[\frac{1}{ t \sqrt{1+t}}-\frac{1}{t} \right]\] this?

  15. anonymous
    • one year ago
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    @idku yes

  16. anonymous
    • one year ago
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    then multiply by conjugate \[\frac{ 1-\sqrt{1+t}}{ t \sqrt{1+t}} \times \frac{ 1+\sqrt{1+t} }{ 1+\sqrt{1+t} }\]

  17. idku
    • one year ago
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    \[\lim_{t \rightarrow 0}\left[\frac{1}{ t \sqrt{1+t}}-\frac{1}{t} \right]\] \[\lim_{t \rightarrow 0}\left[\frac{1}{ t \sqrt{1+t}}-\frac{\sqrt{1+t}}{t \sqrt{1+t}} \right]\] \[\lim_{t \rightarrow 0}\left[\frac{1-\sqrt{1+t}}{ t \sqrt{1+t}} \right]\] I multiplying top and bottom times 1+sqrt(1+t) \[\lim_{t \rightarrow 0}\left[\frac{1-1+t}{ t \sqrt{1+t}+t(1+t)} \right]\] \[\lim_{t \rightarrow 0}\left[\frac{t}{ t (\sqrt{1+t}+(1+t))} \right]\] \[\lim_{t \rightarrow 0}\left[\frac{1}{ \sqrt{1+t}+(1+t)} \right]\]

  18. idku
    • one year ago
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    now plug in t=0 into the limit

  19. idku
    • one year ago
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    AsAAD did this before me, just now noticed:) In any case, this is a full recap then

  20. anonymous
    • one year ago
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    OK! thank you @idku and @ASAAD123 ! Sucks that I can't give 2 best response...

  21. idku
    • one year ago
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    doesn't matter idc

  22. anonymous
    • one year ago
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    @idku ^--^ no problem.

  23. anonymous
    • one year ago
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    correction you missed minus sign \[\lim_{t \rightarrow 0}\frac{ 1-\sqrt{1+t} }{ t \sqrt{1+t} }\times \frac{ 1+\sqrt{1+t} }{ 1+\sqrt{1+t} }\] \[=\lim_{t \rightarrow 0}\frac{ 1-\left( 1+t \right) }{ t \sqrt{1+t}\left( 1+\sqrt{1+t} \right) }\] \[=\lim_{t \rightarrow 0}\frac{ -1 }{ \sqrt{1+t}\left( 1+\sqrt{1+t} \right) }=-\frac{ 1 }{ 2 }\]

  24. anonymous
    • one year ago
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    yea, i noticed it, but thank you!

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