<s>[INST] 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.
 [/INST] $$
\frac 3 x\cdot 2x = 6\text{ since $x$ cancels, and }\frac 1 2\cdot 2x = x\text{ since $2$ cancels.}
$$
Therefore
$$
\frac{\frac{3}{x}-\frac{1}{2}}{x-6}=\frac{\left(\frac{3}{x}-\frac{1}{2}\right)2x}{(x-6)2x} = \frac{6-x}{(x-6)2x} = \frac{-1(x-6)}{(x-6)2x} = \frac{-1}{2x}
$$
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