## anonymous one year ago find the limit as x approaches 1 of abs(x-1)/x-1

1. anonymous

$\lim_{x \rightarrow 1}\frac{ \left| x-1 \right| }{ x-1 }$

2. random231

okay what is the first thing you check for when you solve a limit?

3. dinamix

its 1 or -1

4. anonymous

@random231 you plug it in to see if you get 0/0

5. random231

nope dinamix i dont think you got it!

6. anonymous

@random231 and if so you have to simplify

7. random231

no clara the first thing you have to check whether the limit exists or not!

8. anonymous

$$\large { \lim\limits_{x\to 1}\ \cfrac{|x-1|}{x-1}\implies \begin{cases} \lim\limits_{x\to 1^{\color{red}{ +}}}\ \cfrac{+(x-1)}{x-1} \\ \quad \\ \lim\limits_{x\to 1^{\color{red}{ -}}}\ \cfrac{-(x-1)}{x-1} \end{cases} }$$

9. dinamix

$\lim_{x \rightarrow 1^+} = \frac{ x-1 }{ x-1 } = 1$

10. 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.

11. dinamix

yup @ what i said @random231 its 1 and -1

12. dinamix

13. random231

nope there cant be two values

14. anonymous

@dinamix if thats the case then doesn't the limit not exist? if theyre two different numbers?

15. random231

exactly

16. anonymous

notice the values found for the two one-sided limits they differ thus the two-sided limit of $$\lim\limits_{x\to 1}\ \cfrac{|x-1|}{x-1}\impliedby \textit{does not exist}$$

17. 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