lim [x->-2] [(3x^2+nx+n+3)/(x^2+x-2)]

[scrum]

i'm stuck. This is the last problem in a massive problem set.

a) Find a number "n" such that the following limit exists:

. . . . .lim [x -> -2] [ (3x² + nx + n + 3) / (x² + x - 2)

b) After finding the value of "n", determine the value of the limit.

I have no clue where to begin. I've graphed it plugging random numerals like 2 and 3 and -5 in for n and they just make the asymtotes different but still on -2. I don't know what I should be aiming to cancel or how n can change the limit.
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Edited by stapel -- Reason for edit: Replacing graphic with text.

 

[ilaggoodly]

the issue here is obviously that the bottom is equal to 0, so how would one remove this discontinuity? by simplifying obviously!, start with some factoring, the bottom factors to (x-1)(x+2). so for this to work, we need the top to have one of those things as factors. 3x^2 + nx + n + 3, so we have (3x + )(x + ) and, based on the assumpiton that its possible what is n?

 

[scrum]

wow I feel dumb. I never realized the bottom factored, i was trying to cancel the whole x^2 + x - 2

Someone told me the answer so i got the credit for it and now i'm going it for real. I did the problem knowing the answer but i don't know if I could have done it if i hadn't known n is meant to be 15.

Thanks ever so much for the help though. It helps to see how people got it rather than it being an arbitrary number

 

[galactus]

On the lines of what ilaggoodly said, we know that (x-1) or (x+2) must be a factor. That means a solution is -2 or 1.

If you sub in x=-2, then \(\displaystyle \L\\3(-2)^{2}+n(-2)+n+3=0\)

Now, solve for n. What do you get?.