clara1223
  • clara1223
find the limit as x approaches 1 of abs(x-1)/x-1
Mathematics
katieb
  • katieb
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clara1223
  • clara1223
\[\lim_{x \rightarrow 1}\frac{ \left| x-1 \right| }{ x-1 }\]
random231
  • random231
okay what is the first thing you check for when you solve a limit?
dinamix
  • dinamix
its 1 or -1

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clara1223
  • clara1223
@random231 you plug it in to see if you get 0/0
random231
  • random231
nope dinamix i dont think you got it!
clara1223
  • clara1223
@random231 and if so you have to simplify
random231
  • random231
no clara the first thing you have to check whether the limit exists or not!
jdoe0001
  • jdoe0001
\(\large { \lim\limits_{x\to 1}\ \cfrac{|x-1|}{x-1}\implies \begin{cases} \lim\limits_{x\to 1^{\color{red}{ +}}}\ \cfrac{+(x-1)}{x-1} \\ \quad \\ \lim\limits_{x\to 1^{\color{red}{ -}}}\ \cfrac{-(x-1)}{x-1} \end{cases} }\)
dinamix
  • dinamix
\[\lim_{x \rightarrow 1^+} = \frac{ x-1 }{ x-1 } = 1 \]
random231
  • random231
that is you put in a value immediate to the left of 1 in one case and to the right in the other case. if both of them give the same value then the limit exists.
dinamix
  • dinamix
yup @ what i said @random231 its 1 and -1
dinamix
  • dinamix
the answer
random231
  • random231
nope there cant be two values
clara1223
  • clara1223
@dinamix if thats the case then doesn't the limit not exist? if theyre two different numbers?
random231
  • random231
exactly
jdoe0001
  • jdoe0001
notice the values found for the two one-sided limits they differ thus the two-sided limit of \(\lim\limits_{x\to 1}\ \cfrac{|x-1|}{x-1}\impliedby \textit{does not exist}\)
dinamix
  • dinamix
|dw:1441065218850:dw| this is graph of function now we understand everything i think i am right about my graph draw it @clara1223 , @random231

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