factoring and limits, help!?

Question

Can someone help me factor the following few problems. I'm doing limits but I'm stuck at the basic algebra :\ how shameful. For these I've been asked to solve them analytically (without calculator!) and I can't figure out how to factor the top portion!

1) lim (x^3-6x^2+11x-6) / (x-2)
x>2

2) lim (2x^2-131x+20)/(x-4)
x>4

I hope those are clear illustrations....any help would be soo appreciated!

In the first question,

after doing synthetic division and finding that x-2 is a factor, how did you guys get to this step...

lim (x-2)(x^2-4x+3) / (x-2)

from

lim (x^3 - 6x^2 +11x-6)

I see taking the (x-2) out, but after getting (x^2 - 4x.... I don't know what to do with the +11x and -6... or how they become the rest of the equation???

Answer 1

Long divide if you can. I'm not about to show you how though. That'd be a nightmare. You could also use L'Hopital's rule.

Answer 2

Use long division or synthetic division.

Nonetheless, the numerator of the first problem becomes:
(x^2 - 4x + 3) (x-2)
So after eliminating the (x-2) term and plugging in the value for x, we see the limit is -1.

The second problem doesn't require factoring. When plugging in x = 4, the numerator is positive and the denominator is zero. Clearly, the limit diverges (DNE).

If the second problem was supposed to read:
(2x^2-13x+20)/(x-4), then the numerator factors to:
(2x-5)(x-4).
Clear out the common terms and plug in x=4. The limit is 3.

Answer 3

1.)lim (x^3-6x^2+11x-6) / (x-2)
x>2
lim (x-2)(x^2-4x+3) / (x-2) If you divide out the (x-2) you
x > 2 get

lim (x^2 - 4x + 3) Substitute 2 in for x to get
x > 2

2^2 - 4(2) + 3 = 4 -8 + 3 = -1

2.) lim (2x^2-13x+20)/(x-4)
x>4
I'm assuming that's supposed to be 13x and not 131 b/c that wouldn't work.
lim (x-4)(2x-5)/(x-4) Divide out (x-4)
x>4
lim (2x-5) Substitute 4 in for x
x>4
2(4) - 5 = 3

Answer 4

1. Since the numerator goes to zero at x=2, x=2 is one root, and really all you need to do limit analysis.
2. The numerator does not appear to be factorable.

Answer 5

x^3-6*x^2+11x-6=(x-1)(x-2)(x-3)
so the first one comes to:lim(x-1)(x-3)=-1

I don't know how to do the second.