# You guessed it... a limit question.

• September 4th 2012, 10:09 PM
amillionwinters
You guessed it... a limit question.
I'm having a bit of a difficult time understanding a problem that asks:

Find all values of ${a}$ such that

$\lim_{x\to1}\left(\frac{1}{x-1}-\frac{a}{x^2-1}\right)$

exists and is finite.

I feel that this should be simple, but I am stuck. The following is about as far I can make sense of it:

$\rightarrow \lim_{x\to1}\left(\frac{1}{x-1}-\frac{a}{(x-1)(x+1)}\right)$

$\rightarrow \lim_{x\to1}\left(\frac{x+1}{(x-1)(x+1)}-\frac{a}{(x-1)(x+1)}\right)$

$\rightarrow \lim_{x\to1}\left(\frac{x+1-a}{(x-1)(x+1)}\right)$

Thinking about numbers very close to 1, I believe a = 2; however, I am unable to work through it.
Any help in understanding what I am not seeing would be greatly appreciated.

Take care.
/alan
• September 4th 2012, 10:30 PM
SworD
Re: You guessed it... a limit question.
a=2 will indeed result in a finite limit in this case. If a = 2, then the numerator of your final limit will just be x-1. Then you can reduce the fraction to just 1/(x+1), and limit x --> 1 will exist by plugging in.
• September 5th 2012, 04:20 AM
zaidalyafey
Re: You guessed it... a limit question.
If we use the direct substitution we get $\frac{2-a}{0}$ now for that to have an existing limit

We must have 2-a=0 the a=2

so we have $\lim_{x\to 1}\frac{x-1}{(x-1)(x+1)}=\frac{1}{2}$ so the limit exists

Now we can prove that this is the only solvation by LH rule

so in order for the limit to exist we must have the case $\frac{0}{0}$ so we can apply LH rule

$\lim_{x\to 1}\frac{1}{2x}=\frac{1}{2}$
• September 5th 2012, 02:48 PM
amillionwinters
Re: You guessed it... a limit question.
So, is determining the value of $a$ really just evaluating

$x+1-a=0$

$\rightarrow (1)+1-a=0$

$\rightarrow 2-a=0$

$\rightarrow a=2$

and then plugging in $2$ for $a$ to determine if the limit exists?

I apologize if this a dumb question. I want to be sure I fully understand every limit problem I do, and sometimes the solutions seem simpler to reach than expected.
• September 5th 2012, 03:15 PM
SworD
Re: You guessed it... a limit question.
Yea. If you think about it, at x = 1, the denominator will be 0 and there is nothing you can do about it. So you will have a limit of the form b/0, which is +-infinity, unless b is zero itself, because 0/0 is an indeterminate form. To keep the limit finite, you need to balance the infinitesimal size of the denominator by making the numerator infinitesimal as well, as x --> 1.