Find the limit as x approaches infinity

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Homework Help Overview

The problem involves finding the limit as x approaches infinity of the expression involving square roots: (the square root of x^2 + x minus the square root of x^2 - x). Participants are exploring methods to simplify the expression and evaluate the limit.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the technique of multiplying by the conjugate to simplify the expression. There are questions about how to handle the resulting forms, including indeterminate forms like [inf/inf]. Some participants express confusion about the steps and the involvement of fractions in the process.

Discussion Status

The discussion is ongoing, with participants sharing their attempts and questioning each other's reasoning. Some guidance has been offered regarding the use of the conjugate, but there is no explicit consensus on the correct approach or outcome yet.

Contextual Notes

There are indications of confusion regarding the manipulation of the expression and the resulting forms, with some participants unsure about how to correctly apply the conjugate method. Additionally, there is a mention of potential mistakes in sign handling during calculations.

Jan Hill
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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
 
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Yes, rationalize :-).
 


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.
 


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.
 


"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:
[tex]\lim_{x \to \infty} \left( \sqrt{x^2 + x} - \sqrt{x^2 -x} \right)[/tex]

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


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?
 


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
[tex]\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}}[/tex]
 


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

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