## Limit at -infinity problem

Limits, differentiation, related rates, integration, trig integrals, etc.

### Limit at -infinity problem

$\lim_{x \rightarrow -\infty}\, \left(\sqrt{x^2\, +\, x\, +\, 1}\, +\, x\right)$

The best thing I could think to do here is multiple the numerator and denominator by the conjugate radical sqrt(x^2 + x + 1) - x. But that will lead to a 0 in the denominator once x is factored out. Ialso tried breaking up the quadratic under the radical into factored form, but it has no solutions. Can someone give me a hint?
jsel

Posts: 7
Joined: Mon Oct 11, 2010 11:07 pm

### Re: Limit at -infinity problem

jsel wrote:$\lim_{x \rightarrow -\infty}\, \left(\sqrt{x^2\, +\, x\, +\, 1}\, +\, x\right)$

The best thing I could think to do here is multiple the numerator and denominator by the conjugate radical sqrt(x^2 + x + 1) - x. But that will lead to a 0 in the denominator....

Not actually.

Since $\sqrt{x^2}\, =\, |x|$ and since $x$ will be (very, very) negative, then the square root will "simplify" to $|x|\, =\, -x$.

stapel_eliz

Posts: 1797
Joined: Mon Dec 08, 2008 4:22 pm

### Re: Limit at -infinity problem

Got it, thanks. I forgot about that property when taking the square root of x^2
jsel

Posts: 7
Joined: Mon Oct 11, 2010 11:07 pm

jsel wrote:I forgot about that property when taking the square root of x^2

Everybody does. Expect a trick question using this property on the test.

stapel_eliz

Posts: 1797
Joined: Mon Dec 08, 2008 4:22 pm