Difference between limit at infinity and negative infinity

Question

I know that

\begin{align} \lim\limits_{x\to\infty}(\sqrt {x^2 + x} - \sqrt {x^2 - 1}) & = \lim\limits_{x\to\infty}\frac {x + 1} {\sqrt {x^2 + x} + \sqrt {x^2 - 1}} \\[5 mm] & = \lim\limits_{x\to\infty}\frac {1 + \frac 1 x} {\sqrt {1 + \frac 1 x} + \sqrt {1 - \frac 1 {x^2}}} \\[5 mm] & = \frac 1 2\ . \end{align}

However, why is it that $\lim\limits_{x\to-\infty}(\sqrt {x^2 + x} - \sqrt {x^2 - 1}) = -\frac 1 2\ $? Regardless of whether $x \rightarrow \infty$ or $-\infty\ $, doesn't $\frac 1 x \rightarrow 0\ $? Any intuitions and explanations behind the difference in results will be greatly appreciated :)

Edit

Many thanks to everyone for their input! I know the difference now :)

Answer 1

What you have done to find the limit doesn't work if $ x<0 $.

Let's do this again : \begin{aligned}\lim_{\left|x\right|\to +\infty}{\left(\sqrt{x^{2}+x}-\sqrt{x^{2}-1}\right)}&=\lim_{\left|x\right|\to +\infty}{\frac{x+1}{\sqrt{x^{2}+x}+\sqrt{x^{2}-1}}}\\ &=\lim_{\left|x\right|\to +\infty}{\frac{x+1}{\sqrt{x^{2}\left(1+\frac{1}{x}\right)}+\sqrt{x^{2}\left(1-\frac{1}{x}\right)}}}\\ &=\lim_{\left|x\right|\to +\infty}{\frac{x+1}{\sqrt{x^{2}}\sqrt{1+\frac{1}{x}}+\sqrt{x^{2}}\sqrt{1-\frac{1}{x}}}}\\ &=\lim_{\left|x\right|\to +\infty}\frac{\frac{x}{\left|x\right|}+\frac{1}{\left|x\right|}}{\sqrt{1+\frac{1}{x}}+\sqrt{1-\frac{1}{x}}}\\&=\lim_{\left|x\right|\to +\infty}\frac{\mathrm{sgn}\,{x}+\frac{1}{\left|x\right|}}{\sqrt{1+\frac{1}{x}}+\sqrt{1-\frac{1}{x}}}\end{aligned}

Thus, the limit as $ x\to +\infty $ would be $ \frac{1}{2} $, and the limit as $ x\to -\infty $ would be $ -\frac{1}{2} \cdot $

Answer 2

If $x<0$, $\sqrt {x^2 + x}=-x\sqrt{1+\frac{1}{x}}$.

Answer 3

Note that for $x<0$ when dividing both the numerator and the denominator to $\sqrt{x^2}$, you are actually dividing the numerator by $\sqrt{x^2}=-x$ rather than $x$.

An intuition that why the limits are equal to a real number, is that rather than both the square roots tending to $\infty$, they become closer and closer until their distance tends to a fixed value.