# Limit, Infinity, Radical

credico

## Homework Statement

Evaluate the following limits:

lim sqrt(x^2-3x+1)-x
x->$$\infty$$

lim sqrt(x^2-3x+1)-x
x->-$$\infty$$

2. The attempt at a solution

http://img816.imageshack.us/img816/9995/limitproblem11.jpg [Broken]

We have to enter in the answers online into a program that tells us if we're right or wrong. I got the limit going to positive infinity correct, but I don't know what I am doing wrong going to negative infinity.

Last edited by a moderator:

## Answers and Replies

Homework Helper
Take out a factor of x^2 from the square root and expand $$\sqrt{1+X}$$ as a power series and that should solve the problem for you.

Staff Emeritus
Homework Helper
Gold Member

## Homework Statement

Evaluate the following limits:

limx→-∞ sqrt(x^2-3x+1)-x

2. The attempt at a solution

We have to enter in the answers online into a program that tells us if we're right or wrong. I got the limit going to positive infinity correct, but I don't know what I am doing wrong going to negative infinity.

You did a fine job recognizing that $$\frac{1}{x}=-\,\frac{1}{\sqrt{x^2}}$$ for x < 0.

Once you get to $$\lim_{x\to -\infty}\left(\sqrt{x^2-3x+1}-x\right)=\lim_{x\to -\infty}\left(\frac{-3x+1}{x+\sqrt{x^2-3x+1}}\right)\,,$$ try using L'Hôpital's rule.

Wait! Look at the original! $$\lim_{x\to -\infty}\left(\sqrt{x^2-3x+1}-x\right)$$.

$$\lim_{x\to -\infty}\left(\sqrt{x^2-3x+1}\right)=+\infty$$

and

$$\lim_{x\to -\infty}\left(-x\right)=+\infty$$

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credico
You did a fine job recognizing that $$\frac{1}{x}=-\,\frac{1}{\sqrt{x^2}}$$ for x < 0.

Once you get to $$\lim_{x\to -\infty}\left(\sqrt{x^2-3x+1}-x\right)=\lim_{x\to -\infty}\left(\frac{-3x+1}{x+\sqrt{x^2-3x+1}}\right)\,,$$ try using L'Hôpital's rule.

Wait! Look at the original! $$\lim_{x\to -\infty}\left(\sqrt{x^2-3x+1}-x\right)$$.

$$\lim_{x\to -\infty}\left(\sqrt{x^2-3x+1}\right)=+\infty$$

and

$$\lim_{x\to -\infty}\left(-x\right)=+\infty$$

Haven't learned L'Hopital's rule yet (I know what it is but at this point I think they want us to do it a certain way first).

In regards to what you posted in your edit: You used the Limit Laws to separate the two, but how did you calculate the radical limit? Just by plugging in infinity (or looking at what it does near infinity?) Is that allowed? Are you certain there isn't something that can mislead me by doing this, at least in this case?

Thanks.

Your answer was right by the way. I guess if it's continuous, by definition: lim f(x) as x ->a = lim f(a). It just seems a bit odd since there's that radical floating over top.

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Homework Helper
You did a fine job recognizing that $$\frac{1}{x}=-\,\frac{1}{\sqrt{x^2}}$$ for x < 0.

Once you get to $$\lim_{x\to -\infty}\left(\sqrt{x^2-3x+1}-x\right)=\lim_{x\to -\infty}\left(\frac{-3x+1}{x+\sqrt{x^2-3x+1}}\right)\,,$$ try using L'Hôpital's rule.
Better: Divide both numerator and denominator by x to get
$$\frac{-3+ \frac{1}{x}}{\sqrt{1- \frac{3}{x}+ \frac{1}{x^2}}}$$
Of course, as x goes to infinity, 1/x or $1/x^2$ goes to 0. For the case were x is going to negative infinity, divide by -x (which will be positive) so that you have no problem with it inside the square root.

Wait! Look at the original! $$\lim_{x\to -\infty}\left(\sqrt{x^2-3x+1}-x\right)$$.
$$\lim_{x\to -\infty}\left(\sqrt{x^2-3x+1}\right)=+\infty$$
$$\lim_{x\to -\infty}\left(-x\right)=+\infty$$