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One of the examples in Calculus: A complete course is finding $\lim_{x\to \infty} (\sqrt{x^2+x}-x)$. At first it seems to produce a meningless $\infty-\infty$, but by rationalizing it we eventually come up with $1\over2$. What I don't understand is how it is possible to derive different results depending on the form of the function, since from my understanding two equivalent functions should yield the same result.

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An indeterminate form simply means that we have to do more work to figure out what (if anything) the limit is. It is not, in itself, a result--rather, it is the lack of a result. – Cameron Buie Sep 7 '13 at 15:47
The very expression $\infty-\infty$ produces confusion, and should be avoided. If we imagine $x$ to be very large, it is clear that $\sqrt{x^2+x}$ (and $x$) are very large, but the behaviour of their difference is not obvious. Writing the expression in an equivalent form makes the behaviour for large $x$ clear. As an analogy, if we have a complicated expression $E(x)$, it may not be clear for what $x$ we have $E(x)=0$. Algebraic manipulation of $E(x)$ may make the roots accessible. – André Nicolas Sep 7 '13 at 16:42
up vote 7 down vote accepted

The "non-result" of $\infty - \infty$ upon initial substitution is one of the many indeterminate forms. Obtaining an indeterminate form of a limit is not at all a result: it tells us only that we have to do more work to figure out if and/or what the limit actually is.

So obtaining an indeterminate form, like you found, isn't really meaningless, nor is it a result: it tells us our work has just begun.

Rationalizing the function, as you did in this case, is one technique to help us to actually evaluate a limit, if it exists, in order to determine what the result actually is.

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Perfect, thank you! – Jimmy C Sep 7 '13 at 17:03
You're welcome! – amWhy Sep 7 '13 at 17:06
@amWhy: What nice feedback! +1 – Amzoti Sep 8 '13 at 0:09

As you said, $\infty-\infty$ is meaningless, hence there's no contradiction, the results are not different; in first approach you just fail to present it in a meaningfull way.

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