<s>[INST] Why does this outcome change depending on infinity Why does the outcome of the limit as x approaches infinity of $$\sqrt{x^2+2x}- \sqrt{x^2-2x}$$
which simplifies to
$$\dfrac {4x}{x \left(\sqrt{1 + \frac 2x} + \sqrt {1 - \frac 2x}\right) }= \dfrac {4}{\sqrt{1 + \frac 2x} + \sqrt {1 - \frac 2x}}$$
change depending on whether we take the limit from minus infinity or positive infinity?
If positive, the answer is 2. If negative infinity, the answer is -2??
It seems to me that we get $$ \frac {4}{\sqrt{1} + \sqrt{1}}$$ no matter which side we approach.
 [/INST] The equation $$\sqrt{x^2+2x}=x\sqrt{1+\frac{2}{x}}$$
is true only for $x>0$. The correct equation is $$\sqrt{x^2+2x}=\sqrt{x^2}\sqrt{1+\frac{2}{x}}=|x|\sqrt{1+\frac{2}{x}}$$

The intuition is that for $x>0$ you are subtracting a smaller number from a larger one (i.e. the two square roots); however for $x<0$ you are subtracting a larger number from a smaller one.
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