Understanding the Limit of sqrt(x^2 + 2x -1) + x as x Approaches -infinity

[BluePhone]

Can somebody please explain to me how the limit of this funciton is -1?

http://www.wolframalpha.com/input/?i=lim+%28sqrt%28x^2+%2B+2x+-1%29+%2B+x%29+as+x+approaches+-infinity

I've tried multiplying by the conjugate but that isn't helping since I keep getting negative infinity as the limit if I do that.

 

[QuarkCharmer]

Click show steps on your link. They simply rationalize the function, perform some algebra, and then use L'hopital's rule.

 

[theclock54]

All right, notice that you can multiply by the conjugate. So that's what we do. In essence we're multiplying by one. We multiply the top and bottom by:

square root ((x^2)+2x-1)-x / (x^2)+2x-1)-x

notice it's minus x, that's the conjugate. then when you simplify it you get

(2x-1)/(square root ((x^2)+2x-1)-x

Then you ask what's the highest power? That is x, if you said x^2, that would be wrong because notice it's the square root of x^2, so the highest power is x.

Now you divide everything by the highest power (x), but because you're approaching negative infinity, you have to tack on a (-) negative sign in front of the bottom square root.

now we have on top

(2x/x) - (1/x) now evaluate as x approaches infinity...2x/x the x's cancel and we're left with 2, and 1/x...would be zero.

so on TOP we have 2

now to the bottom, to put something in a square root you have to square it. so on bottom we multiply by (1/x) which gets thrown into square root as dividing everything by x^2, but the -x at the end is not in the square root so we put that over x.

on bottom we have now is -squareroot (x^2/x^2 + 2x/x - 1/x) - x/x
evaluate as x approaches infinity so the x^2 cancel out (leaving 1), and the 2x/x^2 (becomes 0) and the 1/x^2 also becomes zero. and at the end the x/x becomes one

what we're left with now is (-)squareroot of 1 -1, this becomes -1-1 which is -2

and what did we have on top? 2...so now it's 2/-2 which equals -1

that's ur limit..sorry for not using the math functions..i keep messing up on them

 

[eumyang]

I'm assuming that you clicked the "Show steps" button and not understanding the steps shown?

EDIT: Beaten to it.

 

[theclock54]

this is still the limit section, i doubt he knows what l'hopitals rule is. neither do i

 

[BluePhone]

Don't know what Lhopital's rule is.

I followed theclock54's post and understood everything till the part where he says "but because you're approaching negative infinity, you have to tack on a (-) negative sign in front of the bottom square root. ". I see how we get the limit when we do that, but I don't understand why we put a negative sign at the bottom.

Thank you for the help everyone.

 

[SammyS]

It's so handy when the expression can be read right with the post.

lim_(x→-∞) (sqrt(x²+2 x-1)+x) = -1
$\lim_{x\to\,-\infty} (\sqrt{x^2+2 x-1}+x) = -1$

Multiply by $\displaystyle \frac{\sqrt{x^2+2 x-1}-x}{\sqrt{x^2+2 x-1}-x}$

That should give: $\displaystyle \frac{x^2+2 x-1-x^2}{\sqrt{x^2+2 x-1}-x}$

Now, multiply by (1/x)/(1/x)
.

 

[theclock54]

If we were to approach positive infinity, that would mean the x's are all approaching a positive number. But the thing is when I put the x in the square root, I square it right? And if I'm approaching negative infinity, the negative just disappears because it's squared. So we must put a negative sign, to show that the x is negative. It's hard for me to explain, but after doing a few examples, you'll get what's going on.

 

[BluePhone]

I don't really get it but next time I get a similar question I'll just remember to put in the negative sign.

 

[ehild]

A shorter way: $$\sqrt{x^2+2x-1}=\sqrt{(x+1)^2-2}=|x+1|\sqrt{1-\frac{2}{(x+1)^2}}$$

$$\lim_{x \to {-\infty}}|x+1|\sqrt{1-\frac{2}{(x+1)^2}}+x= \lim_{x \to {-\infty}} (|x+1|+x)$$

The last step is for you...

ehild