Expert: Steve Holleran - 5/22/2007

Question
determine the coordinates of any holes for the function:

   x^2 + 4x - 21
y= ----------------
   x^2 - x - 6

Your help is much appreciated. Thank you.  

Answer
Hi Kay,

For rational functions like this one, a "hole" will show up for the x-values which make BOTH the numerator and denominator functions zero.

So, here we can factor:

 y = (x + 7)(x - 3) /  (x - 3)(x + 2)

and you can see that at x = 3, you will get zeros on both top and bottom.

For the y-coordinate, you want to take the limit as x-->3 of the function, and to do this, just cancel the (x -3) factors and take the limit of what remains:

lim,x-->3, y = lim, x-->3, (x + 7) / (x + 2) = 10/5 = 2

So the hole will occur at (3, 2).

I hope this helps out.

Steve Holleran