# Limit is 0/0 ?

1. Sep 30, 2009

### Jules18

Limit is 0/0 ??

1. The problem statement, all variables and given/known data

limit as x approaches -2 of:

(x+2)/(x3+8)

3. The attempt at a solution

I plugged -2 into the eq'n and I got 0 for both the numerator and denomenator, so I thought the answer would be 1.

Apparently it's supposed to be 1/12 ...
is the text wrong?

2. Sep 30, 2009

### DMOC

Re: Limit is 0/0 ??

You need to factor (x^3 - 8).

I remember my PreCalc teacher telling me that if I get 0/0, I can always factor something out.

3. Sep 30, 2009

### Office_Shredder

Staff Emeritus
Re: Limit is 0/0 ??

$$\lim_{x \rightarrow a} \frac{f(x)}{g(x)} = \frac{ \lim_{x \rightarrow a} f(x)}{ \lim_{x \rightarrow a} g(x)}$$

holds only if $\lim_{x \rightarrow a} g(x)$ is not equal to zero. So you can't just split the limit up and get 0/0 (which isn't defined to be 1 anyway).

If x3+8 is equal to 0 at x=-2, it means you can factor an (x-(-2)) = x+2 from it (basic fact about polynomials)

4. Sep 30, 2009

### popo902

Re: Limit is 0/0 ??

did you learn l'hopitals rule?
if you get 0/0 or infinty or undefined, can't you just get the derivative of the top and bottom
so you got 0/0
then f(x)/g(x)
find f'(x)/ g'(x) and keep getting the derivative of top and bottom until you get an actual value when you plug in a, or in this case, -2
am i wrong or...what?

5. Sep 30, 2009

### hanelliot

Re: Limit is 0/0 ??

L'hopital rule is fine of course.. you different top/bottom separately and get 1/3x^2, which is 1/12 if you plug in -2.

6. Oct 1, 2009

### DrMath

Re: Limit is 0/0 ??

if you get "0/0" - you should always use L'Hopital Rule.
Note that "0/1" or "1/0" results you shouldn't use... :)

7. Oct 1, 2009

### Staff: Mentor

Re: Limit is 0/0 ??

L'Hopital's Rule is overkill for this problem, and is therefore not needed. The denominator can be factored into (x + 2) times a quadratic. The sum or difference of two cubes can be factored as follows:

a3 + b3 = (a + b)(a2 - ab + b2)
a3 - b3 = (a - b)(a2 + ab + b2)

8. Oct 1, 2009

### DMOC

Re: Limit is 0/0 ??

^
Agreed.

9. Oct 1, 2009

### HallsofIvy

Staff Emeritus
Re: Limit is 0/0 ??

In fact, any time a polynomial, P(x), say, equals 0 for x= a, it must have x- a as a factor. You can use "long division" or "synthetic division" to determine the other factor.

10. Oct 1, 2009

### Staff: Mentor

Re: Limit is 0/0 ??

I don't know about that. L'Hopital's rule is more general (applies to non-polynomials) and differentiation is easier than factoring. Given the choice between an easy general solution and a difficult specialized solution I will pick the easy general approach every time.

Last edited: Oct 1, 2009
11. Oct 1, 2009

### Staff: Mentor

Re: Limit is 0/0 ??

Notice that I said "for this problem." Also, factoring is almost always taught before differentiation, so if the OP hasn't been exposed to differentiation yet, then L'Hopital's Rule would be out of the question, regardless of its generality.