<s>[INST] How does he get a perfect swap numerator and denominator. I'm going through a exercise, in which all the answers are given, but the tutor makes a step and I can't follow at all. A massive jump with no explanation.
Here is the question:
$\lim_{x \to 2} \frac{\frac{1}{2}-\frac{1}{x}}{x-2}$
He then simplifies:
$ \frac{x-2}{2x(x-2)}$
He said he multiplied the entire equation by 2x
How does he know 2x swaps the denominator upto the numerator 
Before he gave the simplification I spent perhaps 20 minutes trying to figure something out, and failed, and then he just jumps this massive step
He simplifies because we are finding limits.
Thanks
Joseph G.
 [/INST] Note that
\begin{align*}
\frac{\frac{1}{2}-\frac{1}{x}}{x-2} &= \frac{\frac{x-2}{2x}}{x-2} \\\ &= \frac{2x \cdot \frac{x-2}{2x}}{2x \cdot (x-2)} \qquad \Bigl[\small\text{Multiplying numerator and denominator by} \ 2x \Bigr] \\\ &= \frac{(x-2)}{2x \cdot (x-2)} \\\ &= \frac{1}{2x}
\end{align*}
</s>