Solve Limit Question: Get Help Now

[nesan]

Limit question, please help!?

Homework Statement

lim as x -> -1 of the function

(108 (x^2 + 2x)(x + 1)^3) / ((x^3 + 1)^3 (x - 1))

The Attempt at a Solution

Tried in like 10 different ways, came no where close to the answer. I just need someone to point me in the right direction, thank you. :)

 

[SammyS]

Homework Statement

lim as x -> -1 of the function

(108 (x^2 + 2x)(x + 1)^3) / ((x^3 + 1)^3 (x - 1))

The Attempt at a Solution

Tried in like 10 different ways, came no where close to the answer. I just need someone to point me in the right direction, thank you. :)

What were some of the ways you tried?

Show what you did so we can help you.

 

[nesan]

Direct substitution gives a 0 / 0.

First, I tried expanding the (x + 1) ^3

to

(x + 1) (x^2 - x + 1)

and (x^3 + 1)^3

to

(x^3 + 1) ((x^3)^2 - x^3 + 1)

That didn't help.

Second, I tried distributing the 108 and everything else on the top into one polynomial, which ended up in a mess.

I'm sure there's an easier way, I just can't seem to figure it out. :(

I tried to find ways to cancel out one term from the bottom, but it always ends up with the opposite sign in the top.

If someone can point me in the right direction, I'll try to get the limit. :)

 

[alanlu]

What is the degree of the polynomial in the numerator? What about the denominator?

Edit: use SammyS's advice. I read the problem wrong.

 

[nesan]

I only did the numerator, which has a degree of 5.

I only did the numerator because for some reason I though I could combine all and factor it in a different way, but it ended up in a mess and I lost track of everything. I don't think it is suppose to be this complex.

For example, it has a (x - 1) at the bottom but the closest I can come to matching it is (x + 1) at the top. :(

 

[SammyS]

...

First, I tried expanding the (x + 1) ^3

to

(x + 1) (x^2 - x + 1)

and (x^3 + 1)^3

to

(x^3 + 1) ((x^3)^2 - x^3 + 1)

Well, those are incorrect.
(x + 1) (x² - x + 1) = x³ + 1, not (x+1)³ .

(x + 1)³ = x³ + 3x² + 3x + 1

That didn't help.

Second, I tried distributing the 108 and everything else on the top into one polynomial, which ended up in a mess.

I'm sure there's an easier way, I just can't seem to figure it out. :(

I tried to find ways to cancel out one term from the bottom, but it always ends up with the opposite sign in the top.

If someone can point me in the right direction, I'll try to get the limit. :)

Factor the x³+1 in the denominator.

x³+1 = (x+1)(x² - x + 1)​

Since the x³+1 in the denominator is cubed, that should give a factor in the denominator which cancels with (x+1)³ in the numerator.

 

[nesan]

Well, those are incorrect.
(x + 1) (x² - x + 1) = x³ + 1, not (x+1)³ .

(x + 1)³ = x³ + 3x² + 3x + 1



Factor the x³+1 in the denominator.

x³+1 = (x+1)(x² - x + 1)​

Since the x³+1 in the denominator is cubed, that should give a factor in the denominator which cancels with (x+1)³ in the numerator.

Sweet, got it. Thank you very much. :)