Can someone tell me the limit of...

[K9-47G]

Can someone tell me the limit of (x/(2x-2))-(1/((x^2)-1)) as x approaches 1.

[TD]

Can someone tell me the limit of (x/(2x-2))-(1/((x^2)-1)) as x approaches 1.

Rewrite the fractions, expand the second one, add them and simplify.

 

[math]

\begin{gathered}

\frac{x}

{{2x - 2}} - \frac{1}

{{x^2 - 1}} \hfill \\

\frac{x}

{{2\left( {x - 1} \right)}} - \frac{1}

{{\left( {x - 1} \right)\left( {x + 1} \right)}} \hfill \\

\frac{x}

{{2\left( {x - 1} \right)}} - \left( {\frac{1}

{{2\left( {x - 1} \right)}} - \frac{1}

{{2\left( {x + 1} \right)}}} \right) \hfill \\

\frac{{x + 2}}

{{2\left( {x + 1} \right)}} \hfill \\

\end{gathered}

[/math]

 

Now fill in x = 1

[K9-47G]

Thank you so much!

[TD]

No problem

[BobbyJoeCool]

[math]\frac{x}{2(x-1)}-\frac{1}{(x-1)(x+1)}[/math]

 

[math]\frac{x}{2(x-1)}-(\frac{1}{2(x-1)}-\frac{1}{2(x+1)})[/math]

 

How do you figure that?

[Dave]

Haven't checked it, but probably partial fractions.

[TD]

Yes, you can split using partial fractions or by "playing with fractions" like this:

 

[BobbyJoeCool]

Yes' date=' you can split using partial fractions or by "playing with fractions" like this:

 

[img']http://ld.livedelivery.com/F/2157642/Image.gif[/img]

 

I don't see how any of the second half of these work. It seems to me to not work. Is this something I'd learn after Calculus I?

[TD]

Well normally you'd split by doing partial fractions but what I did doesn't require any advanced calculus. I added and substracted same things in the nominator (e.g. replaced "1" by "1+x-x" in the first step) and then split the fraction to simplify etc...