(3x-12)/(x^2-5x) * 7x/(4-x)
[My multiply sign is *.]
Notice the x/x and (x-4)/(x-4).
x/x = 1 for all real numbers EXCEPT 0, where it's UNDEFINED.
(x-4)/(x-4) = 1 everywhere EXCEPT 4, where it's UNDEFINED.
These are called "holes". We'll exclude them by stating x <> 0,4.
[My not-equal sign is <>.] Then we can cancel them:
-21/(x-5), x <> 0,4,5.
x <> 5 b/c it would cause division by zero.
In fact, x = 5 is a vertical asymptote.
The horizontal asymptote is y = 0.