# How to rewrite limit to prove?

1. Feb 10, 2008

### ianq

Hi,
I'm taking calculus I in college right now and for some reason we stated with limits...We're giving the following limit (sorry, I don't know how to work the board's code to make it look pretty):

lim (x->3) is (x+6) / (x^4 - 4x^3 + x^2 + x + 6) = -1

The prof suggested we rewrite lim (x->3) is (x+6) / (x^4 - 4x^3 + x^2 + x + 6) + 1 = 0 in the form (x-3)g(x) and find g(x). Any idea what g(x) would be and how to find it?

2. Feb 10, 2008

### d_leet

Did you ever think about just plugging in 3 for x and seeing what happens?

3. Feb 10, 2008

### ianq

Yep, I did. I get -1 = -1. But considering it's a class exercise and the prof wants us to rewrite it as (x-3)g(x) I'm clueless...

4. Feb 10, 2008

### HallsofIvy

Staff Emeritus
so do the algebra! What is
$$\frac{x+6}{x^4- 4x^3+ x^2+ x+ 6}+ 1$$?

I assume you know that is the same as
$$\frac{x+6}{x^4- 4x^3+ x^2+ x+ 6}+ \frac{x^4- 4x^3+ x^2+ x+ 6}{x^4- 4x^3+ x^2+ x+ 6}$$

Add and try to factor the numerator. If you already know one factor, that should be easy!