<s>[INST] Evaluating $\lim \limits_{x\to -\infty} x + \sqrt{x^2+2x}$ which step is wrong? So I'm trying to evaluate $\lim \limits_{x\to -\infty} x + \sqrt{x^2+2x}$
These are my steps:
I first rationalize the expression (square root trick) -
$$\lim \limits_{x\to -\infty} \frac{-2x}{x - \sqrt{x^2+2x}}$$
Then I simply divide by $x$ so
$$\lim \limits_{x\to -\infty} \frac{-2}{1 - \sqrt{1+\frac{1}{2x}}}$$
Then I get the following by evaluating the limit
$$\frac{-2}{1 - \sqrt{1}}$$
which then evaluates to $0$ in the denominator.
Would really appreciate some help in understanding what I'm doing wrong here.
Thanks in advance!
 [/INST] What you got wrong is in the step "I simply divide by $x$", where you should get
$$
\frac{-2x}{x - \sqrt{x^2+2x}}=
\frac{-2}{1-\frac{1}{x}\sqrt{x^2+2x}}
=\frac{-2}{1{\color{red}+}\sqrt{\frac{x^2+2x}{x^2}}}
=\frac{-2}{1{\color{red}+}\sqrt{1+\frac{2}{x}}}
$$
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