I'm working on problem set 1, problem is to show the type of discontinuity of (x-2)/(x^2-4)

It's apparent just by looking that the function is not defined at x=2, and that the discontinuity is removeable.

My problem is in working through a proof. Using the quotient rule I can get to -(x-2)^2/(x^2-4)^4, but I can't reduce it from there. I know the answer is there, brain fade is stopping me seeing it.

Question

I'm working on problem set 1, problem is to show the type of discontinuity of (x-2)/(x^2-4)

It's apparent just by looking that the function is not defined at x=2, and that the discontinuity is removeable.

My problem is in working through a proof. Using the quotient rule I can get to -(x-2)^2/(x^2-4)^4, but I can't reduce it from there. I know the answer is there, brain fade is stopping me seeing it.

noelgreco · Answer

The denominator if your derivative should be ^2, not ^4.  That factors to (x-2)^2(x+2)^2

anonymous · Answer

Opps that's a typo. Derivative is:

$$-\frac{ (x-2)^{2} }{ (x ^{2}-4)^{2} }$$

and I'm trying to reduce it to

$$-\frac{ 1 }{ (x-2)^2 }$$

anonymous · Answer

$$-\frac{(x-2)^2}{(x^2-4)^2}=-\frac{(x-2)^2}{((x-2)*(x+2))^2}=\frac{(x-2)^2}{(x-2)^2*(x+2)^2}$$

anonymous · Answer

Thanks! It now seems so obvious...