Arithmetic Error in Calculation of the Limit of a Given Function I consider a function $f(x)$ which is equal to $\dfrac{\frac{3}{x}-\frac{1}{2}}{x-6}$
While trying to evaluate the $\lim_{x \to 6} f(x)$
It is true that $\dfrac{\frac{3}{x}-\frac{1}{2}}{x-6} = \dfrac{\frac{3}{x}}{x-6}-\dfrac{\frac{1}{2}}{x-6}$
It is also true that $\dfrac{a}{\frac{b}{c}} = \dfrac{a\cdot c}{b}$
I apply this property to $f(x)$ and come to the conclusion that $\dfrac{\frac{3}{x}-\frac{1}{2}}{x-6} = \dfrac{\frac{3}{x}}{x-6}-\dfrac{\frac{1}{2}}{x-6} = \dfrac{3x-18}{x}-\dfrac{x-6}{2}$, which is of course incorrect.
Additionally, I believe I'm taking an improper approach to solving this limit.

Note: $$\frac a{\frac bc} = \frac{ac}b$$
You arrive at only $ac$.
Also, $$\dfrac{\frac ab}{c} = \frac{a}{bc}$$
So $$ \dfrac{\frac{3}{x}-\frac{1}{2}}{x-6} = \dfrac{\frac{3}{x}}{x-6}-\dfrac{\frac{1}{2}}{x-6} = \frac{3}{x(x-6)} - \frac 1{2(x-6)} = \frac{2\cdot 3}{2x(x-6)}-\frac{x}{2x(x-6)} = \dfrac{6-x}{2x(x-6)}$$ 
$$= \frac{-(x-6)}{2x(x-6)} = \frac{-1}{2x}$$
