What is the Limit to Infinity of a Rational Function?

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Homework Help Overview

The discussion revolves around evaluating the limit of a rational function as x approaches infinity, specifically the expression lim_x->∞ of [√x + 2|x|] / [1 + x].

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to simplify the limit by dividing the numerator and denominator by the highest power of x. They express uncertainty about the correctness of their approach, particularly regarding the presence of a \pm sign in their answer.

Discussion Status

Some participants question the use of the \pm sign in the context of x approaching positive infinity, suggesting a need for clarification. There is a brief exchange confirming the correctness of the approach, but no consensus is reached on the final interpretation of the limit.

Contextual Notes

The discussion includes a reference to the behavior of x as it approaches positive infinity, indicating that assumptions about the sign of x are being examined.

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Homework Statement



lim_x->∞ of [√x + 2|x|] / [1 + x]

Homework Equations


lim_x->∞ of 1/[x^n] = 0


The Attempt at a Solution


Hi everyone,

Here's what I've done so far:

I divided top and bottom by the highest power of x, i.e. x.
So:
[x^-1/2 +/- 2] / [1/x + 1]

And taking the limit of x to ∞ of this:
[0 +/- 2] / [0 + 1] = +/- 2

Is this correct?

Thanks for any help.
 
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Why is there a [itex]\pm[/itex] in your answer?...Surely you recognize that [itex]x[/itex] is positive as it approaches [itex]+\infty[/itex]?
 
gabbagabbahey said:
Why is there a [itex]\pm[/itex] in your answer?...Surely you recognize that [itex]x[/itex] is positive as it approaches [itex]+\infty[/itex]?

Thanks, I didn't think about that. But is it correct apart from that?
 
Yes.
 

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