Expert: Alon Mandes - 9/25/2008

Question
Hi
I have a limit that looks as such:
lim x -> -3
(x^2-x+12) / (x+3).
1. When I sub in the -3 for x, I get the limit as 24/0.
2. When I redo the limit as one sided, I get a + infinity / - infinity, meaning the limit does not exist.
3. If I factor out the equation, the (x + 3 ) will cancel out and I get an answer of -7.
Which of these 3 methods is the appropriate and correct method?
Thank you.

Answer
Well Stephanie , as you can see the function (x^2-x-12)/(x+3) has
singularity in x=-3. Singularity means that the function value in
this point jumps to ∞. So instead of plugging -3 in the function
another approach is required...Let's see if this function can be
factorized : (x^2-x-12)/(x+3)=(x+3)(x-4)/(x+3)=x-4. So, this means
that there is no singularity, the function doesn't jump to  ∞, it
has a finite value of -7. But, is this function continues in this
point ???
Before answering that, we will sum what we did so far .
1st we chick if there is singularity in the given point or not.
2nd we try to factorize the equation to see of we get rid of the
singularity. This is the way of calculating the limit.
3rd we will chick continuesity by taking limit from left & from
right.
Right now we proved that Lim{x->-3} (x^2-x-12)/(x+3)=-7.
The function x-4 is continues meaning Lim from the right = Lim from the left.

Alon.