# Homework Help: More limits problems

1. Sep 22, 2008

### Melawrghk

1. The problem statement, all variables and given/known data
1. limx->-inf (sqrt(x2+6x-1) + x)
2. Let g(x) = x10/9, find limh->0 g((209+h) - g(209))/h

2. Relevant equations
None that I know of.

3. The attempt at a solution
1. Well, first I put the -infinty in and I think it's an indetermination because it's inf-inf. So I decided to rationalize the equation and got:
6x-1
------------
sqrt(x2+6x-1) -x)
I figured the limit of the denominator has to be +infinity, limit of the top is -infinity, which would result in getting a -infinity. But the answer says it's -3.

2. I substituted and got this:
(209+h)10/9 - 209
-----------------------
h

Which really gets me nowhere because it is still unclear at to what to do with the first part of the numerator. I'm only allowed to solve this with regular limit rules, nothign fancy...

2. Sep 22, 2008

### tiny-tim

Hi Melawrghk!

(have an infinity: ∞ and a square-root: √ )

Hint: 1. can you solve limx->-∞ (√(x2+6x+9) + x)?

2. what is (1 + x)10/9 ?

3. Sep 22, 2008

### Melawrghk

Nope :) Because that is essentially what I'm asking. It turns out to be ∞-∞ and I'm not sure if the infinites are equal. Plus, it definitely doesn't equal -3...

Honestly, I have no clue. 9th root of (1+x)10?

4. Sep 22, 2008

### tiny-tim

Yup

(what is √(x2+6x+9) ?)

Try again!
ok … try: what is (1 + x)10 if x is very small?

(1 + x)5 ?

(1 + x)10/9 ?

5. Sep 22, 2008

### Dick

For first one try this. sqrt(x^2+6x-1)=sqrt(x^2*(1+6/x-1/x^2))=|x|sqrt(1+6/x-1/x^2). Apply that to your rationalized form.

6. Sep 22, 2008

### Melawrghk

OOH I'm dumb! :D Okay, I get it now. Can't believe I overlooked that...
EDIT: Wait, no. I get this one, but how would I do it with mine? Mine doesn't factor nicely...
It's 1whatever. But if I use that tactic, won't I just end up with 0/0 again? Because if I eliminate the "h" in the numerator, I'll be left with two equal but opposite terms... Sorry, I still don't get this one.

Thanks Dick, I'll try that.

7. Sep 22, 2008

### tiny-tim

Use x2 + 6x - 1 = (x2 + 6x + 9)(1 - 10/(x2 + 6x - 1))
(1 + x)10 = 1 +10x +15x2 + …

so, for very small x, (1 + x)10 is approximately 1 +10x.

8. Sep 22, 2008

### Melawrghk

Where'd you get that?...

Yeah.. binomial expansion. But that's for a nice power. 10/9 is ANYTHING but nice.
Maybe if I do the 9th root of 209+h to the power of 10 that would help. I'll try.

PS. Thanks Dick, it worked :)