Find the limit as x approaches infinity

[Jan Hill]

Homework Statement

Find the limit as x approaches infinity of (the square root of x^2 + x minus the square root of x^2 -x)

Homework Equations

The Attempt at a Solution

I do not know how to simplify the expression. I know that plugging in x = inifinity would be wrong. How about multiplying by the conjugate i.e. (the square root of x^2 +x plus the sqaure root of x^ 2 - x

 

[╔(σ_σ)╝]

Yes, rationalize :-).

 

[Mark44]

After you multiply by 1 in the form of sqrt(x^2 + x) + sqrt(x^2 - x), you will get another indeterminate form, [inf/inf]. You can deal with this one by factoring x^2 out of each term in the two radicals, and bringing it out of the radicals as x.

 

[Jan Hill]

When I work this out, I get zero in the numerator. I don't know how to get the indeterminate form, [inf/inf] that you got. How do you get that?

I end up with 2x^2 - 2x^2 in the numerator when I myultiply by the conjugate.

 

[eumyang]

"Find the limit as x approaches infinity of (the square root of x^2 + x minus the square root of x^2 -x)"

I must be missing something. Why are we talking about fractions? This is how I'm reading the problem:
\lim_{x \to \infty} \left( \sqrt{x^2 + x} - \sqrt{x^2 -x} \right)

Or is the idea to multiply this by a fraction where the numerator and denominator is the conjugate?

 

[Jan Hill]

You're right...I don't know how I got fractions involved.


Mulitplying by the conjugate, I end up with the limit as x---> infinity of 2x

Is this right?

 

[Mark44]

The fractions got involved because of the suggestion to multiply by the conjugate. Since you can't just multiply by the conjugate, but instead need to multiply by 1 in the form of the conjugate over itself, you end up with a fraction.

IOW, the original expression can be written as
\sqrt{x^2 + x} - \sqrt{x^2 - x} \cdot \frac{\sqrt{x^2 + x} + \sqrt{x^2 - x}}{\sqrt{x^2 + x} + \sqrt{x^2 - x}}

 

[Mark44]

When I work this out, I get zero in the numerator. I don't know how to get the indeterminate form, [inf/inf] that you got. How do you get that?

You have made a mistake. You should not get zero in the numerator. Check your signs.

I end up with 2x^2 - 2x^2 in the numerator when I myultiply by the conjugate.