1. ## Limits

Hi everyone!

I started revision today and im stuck on a limit question. I have the solution but I still have no clue of how to solve it. maybe another simple example may help.

Question 1.
Evaluate the limit of $\frac{(x+1)}{(|x|-1)}$ as $x\rightarrow-1$

If $x=-1$ then get $\frac{-1+1}{1-1}=\frac{0}{0}!$

Therefore let $x=-1-\in$ then $\in\rightarrow0$ in limit $(\in >0)$

i.e. $\frac{-1-\in+1}{(1+\in)-1} = \frac{-\in}{\in}=-1$

$x=-1-\in=-(1+\in)$
$|-1-\in|=(1+\in)$

If let $x=-1+\in=-(1-\in)$ and $|-1+\in|=1-\in$ $(\in>0)$

Therefore
$\frac{x+1}{|x|-1}$ with $x=-1+\in$ is $\frac{-1+\in+1}{1-\in-1}=\frac{\in}{-\in}=-1$ same answer as before

Thats the answer above. I think they use two solutions but both ways are right. But I don't get why you have to put in $x=-1-\in$. Hopefully someone can help.

Thank you

Hi everyone!

I started revision today and im stuck on a limit question. I have the solution but I still have no clue of how to solve it. maybe another simple example may help.

Question 1.
Evaluate the limit of $\frac{(x+1)}{(|x|-1)}$ as $x\rightarrow-1$

If $x=-1$ then get $\frac{-1+1}{1-1}=\frac{0}{0}!$

Therefore let $x=-1-\in$ then $\in\rightarrow0$ in limit $(\in >0)$

i.e. $\frac{-1-\in+1}{(1+\in)-1} = \frac{-\in}{\in}=-1$

$x=-1-\in=-(1+\in)$
$|-1-\in|=(1+\in)$

If let $x=-1+\in=-(1-\in)$ and $|-1+\in|=1-\in$ $(\in>0)$

Therefore
$\frac{x+1}{|x|-1}$ with $x=-1+\in$ is $\frac{-1+\in+1}{1-\in-1}=\frac{\in}{-\in}=-1$ same answer as before

Thats the answer above. I think they use two solutions but both ways are right. But I don't get why you have to put in $x=-1-\in$. Hopefully someone can help.

Thank you

You use $x=1\pm \epsilon, \epsilon > 0$ as you want to look at the behaviour of the expression near $x=-1$, and you need to handle the $\pm \epsilon$ cases seperatly to control the behaviour of $|1 \pm \epsilon|$ while looking at the limit from above and below.

RonL

Hi everyone!

I started revision today and im stuck on a limit question. I have the solution but I still have no clue of how to solve it. maybe another simple example may help.

Question 1.
Evaluate the limit of $\frac{(x+1)}{(|x|-1)}$ as $x\rightarrow-1$

If $x=-1$ then get $\frac{-1+1}{1-1}=\frac{0}{0}!$

Therefore let $x=-1-\in$ then $\in\rightarrow0$ in limit $(\in >0)$

i.e. $\frac{-1-\in+1}{(1+\in)-1} = \frac{-\in}{\in}=-1$

$x=-1-\in=-(1+\in)$
$|-1-\in|=(1+\in)$

If let $x=-1+\in=-(1-\in)$ and $|-1+\in|=1-\in$ $(\in>0)$

Therefore
$\frac{x+1}{|x|-1}$ with $x=-1+\in$ is $\frac{-1+\in+1}{1-\in-1}=\frac{\in}{-\in}=-1$ same answer as before

Thats the answer above. I think they use two solutions but both ways are right. But I don't get why you have to put in $x=-1-\in$. Hopefully someone can help.

Thank you