If the exponent of the first term in the numerator (upper equation) and the exponent in the first term of the denominator are the same, then the horizontal asymptote is at y = the ratio of the two.

  Ex.    4x² + 6 

           2x² - 3

 So the asymptote is at y = 4/2  which is reduced to y = 2.

If the exponent of first term in the numerator is smaller than the exponent in the first term of the denominator, then the horizontal asymptote is at y = 1.

  Ex.   x + 6

          x² - 3

The exponent on the x of the numerator is 1 and the one on the denominator is 2 so the horizontal asymptote is at y = 1.

If the exponent on the first term of the numerator is greater than the exponent on the first term of the denominator, then you have an oblique asymptote on the line made by dividing the denominator into the numerator and discarding the remainder.

   Ex. 

 x² + 3x + 9

     x – 4                use either synthetic or long division to divide the numerator by the denominator

this gives you x + 7 with a remainder of -19. Disregarding the -19, the oblique asymptote will be

the line y = x + 7.