Question

Give an example of a polynomial that has the asymptotes of x=5 and y=0.5

Answer 1

(x²+4x+3)/2(x²-10x+25)

the horizontal asymptote when the numerator and the denominator have the same degree (in this case, both of a degree of 2) is ration of the coefficients of the numerator and denominator. In this case, the coefficient for numerator x² is 1, and the coefficient for the denominator 2x² is 2, so the horizontal asymptote is y=1/2=0.5

the vertical asymptote is the x value. the denominator cannot be zero, if x²-10x+25=0, x would be 5, so the vertical asymptote is x=5 

this is just one example. There can be others:
(2x²+5x+2)/[(4x-7)(x-5)]  for another example, but this example has a second vertical asymptote 4x-7=0 =>x=7/4