anonymous
  • anonymous
lim t->0 [1/(t√(1+t))-(1/t)]
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
\[\lim_{x \rightarrow 16} (\frac{ 1 }{ t \sqrt{1+t} } - \frac{ 1 }{ t })\]
dinamix
  • dinamix
(1-(17)^1/2 /(17)^1/2) 1/16
Michele_Laino
  • Michele_Laino
hint: your function, is continue at t=16

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anonymous
  • anonymous
*as t approaches 0... sorry
Michele_Laino
  • Michele_Laino
you have to replace t with 16 only
dinamix
  • dinamix
its \[-\]
dinamix
  • dinamix
when t---->0
anonymous
  • anonymous
the answer is -1/2
anonymous
  • anonymous
yes, the answer is -1/2 @vvvbb
anonymous
  • anonymous
yes... so how do I get there?
anonymous
  • anonymous
|dw:1440949656458:dw| choose common denominator and simplify
anonymous
  • anonymous
then you get\[\frac{ 1-\sqrt{1+t} }{ t \sqrt{1+t}}\]
anonymous
  • anonymous
|dw:1440949891651:dw|
idku
  • idku
\[\lim_{t \rightarrow 0}\left[\frac{1}{ t \sqrt{1+t}}-\frac{1}{t} \right]\] this?
anonymous
  • anonymous
@idku yes
anonymous
  • anonymous
then multiply by conjugate \[\frac{ 1-\sqrt{1+t}}{ t \sqrt{1+t}} \times \frac{ 1+\sqrt{1+t} }{ 1+\sqrt{1+t} }\]
idku
  • idku
\[\lim_{t \rightarrow 0}\left[\frac{1}{ t \sqrt{1+t}}-\frac{1}{t} \right]\] \[\lim_{t \rightarrow 0}\left[\frac{1}{ t \sqrt{1+t}}-\frac{\sqrt{1+t}}{t \sqrt{1+t}} \right]\] \[\lim_{t \rightarrow 0}\left[\frac{1-\sqrt{1+t}}{ t \sqrt{1+t}} \right]\] I multiplying top and bottom times 1+sqrt(1+t) \[\lim_{t \rightarrow 0}\left[\frac{1-1+t}{ t \sqrt{1+t}+t(1+t)} \right]\] \[\lim_{t \rightarrow 0}\left[\frac{t}{ t (\sqrt{1+t}+(1+t))} \right]\] \[\lim_{t \rightarrow 0}\left[\frac{1}{ \sqrt{1+t}+(1+t)} \right]\]
idku
  • idku
now plug in t=0 into the limit
idku
  • idku
AsAAD did this before me, just now noticed:) In any case, this is a full recap then
anonymous
  • anonymous
OK! thank you @idku and @ASAAD123 ! Sucks that I can't give 2 best response...
idku
  • idku
doesn't matter idc
anonymous
  • anonymous
@idku ^--^ no problem.
anonymous
  • anonymous
correction you missed minus sign \[\lim_{t \rightarrow 0}\frac{ 1-\sqrt{1+t} }{ t \sqrt{1+t} }\times \frac{ 1+\sqrt{1+t} }{ 1+\sqrt{1+t} }\] \[=\lim_{t \rightarrow 0}\frac{ 1-\left( 1+t \right) }{ t \sqrt{1+t}\left( 1+\sqrt{1+t} \right) }\] \[=\lim_{t \rightarrow 0}\frac{ -1 }{ \sqrt{1+t}\left( 1+\sqrt{1+t} \right) }=-\frac{ 1 }{ 2 }\]
anonymous
  • anonymous
yea, i noticed it, but thank you!

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