How to rewrite limit to prove?

[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?

 

[d_leet]

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

 

[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...

 

[HallsofIvy]

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?

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!