Question

find the limit as x approaches 1 of abs(x-1)/x-1

Answer 1

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

Answer 2

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

Answer 3

its 1 or -1

Answer 4

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

Answer 5

nope dinamix i dont think you got it!

Answer 6

@random231 and if so you have to simplify

Answer 7

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

Answer 8

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

Answer 9

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

Answer 10

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.

Answer 11

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

Answer 12

the answer

Answer 13

nope there cant be two values

Answer 14

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

Answer 15

exactly

Answer 16

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

Answer 17

this is graph of function now we understand everything i think i am right about my graph draw it @clara1223 , @random231