Question 811469
Imagine the degree of numerator is equal to that of the denominator, and that one of the expression or factors in the denominator is (x-5).


Try something simple, {{{(x-5)(x-1/2)/(x-5)}}}.  Seeing the way the denominator is chosen, x=5 will be a vertical asymptote.  This will NOT give the horizontal asymptote wanted; so try a change.


{{{highlight(x(x-5)/((x-5)(2x)))}}}
Still the same x=5 vertical asymptote.  Also there is now x=0 vertical asymptote. Note what would happen as x is increased without bound...  The x in numerator and denominator would cancel as 1, and the x-5 in numerator and denominator would cancel as 1, and the expression would approach (1/2), being {{{y=1/2}}} as horizontal asymptote.