Equation: x^2 - 64 / ((3x+4)(x-5))
The horizontal asymptote is 1/3. I graphed this and for very large values of x, the graph equals appromixmately .354 and a little up. How is 1/3 an asymptote then? My books ays to make x equal a very large positive number, and then it will be (x^2)/(3x^(2)), which is 1/3. Am I looking at the graph wrongly?
Another way to think about this is to think of how "powerful" each term in a polynomial is. In the numerator, x^2 is more "powerful" than -64, since x^2 increases more rapidly (... well, -64 doesn't increase or decrease). So when x is large*, the x^2 term will dominate.
Likewise for the denominator. For x "sufficiently large", the 3x^2 term will dominate, so your fraction will be - roughly:
, and you can see that this is 3.