How can evaluating the limit of function give a different result after rationalizing it?

Question

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.

Answer 1

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.

Answer 2

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.