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.

$$\sqrt{x^2} = |x|$$
So when $x$ tends to $- \infty$, $\sqrt{x^2} = -x$.
Hence then answer would be:
$$\frac{4}{-2} = -2$$
