Limits of Functions

When we speak of N = lim x -> a, f(x), this reads as N is the limit of the function f(x) as x approaches a from either side. It's like trying to go somewhere forever, but never actually getting there. This then forms an asymptote, which is for this example an asymptote at x = a.

Methods..
For the function f(x) = (3x^2+2x+1)/(x^2+4x+5), with lim x -> inf, you can do one of two things.
One, you can divide by the greatest power of x, which is x^2 through the equation, or you can factor out the x as much as you can.
By taking the first method, you will obtain lim x -> inf [(3+2/x+1/x^2)/(1+4/x+5/x^2)]. Substituting in infinity for x blows up a lot of those fractions yielding 3/1 = 3.
The other method is that you factor as much as you can.. obtaining something ugly. So in some cases it's better to just do the division by the highest power of x.
For f(x) = (x^2+x-6)/(x-2), with lim x -> 2, factoring is fine.
f(x) = (x-2)(x+3)/(x-2). f(x) = x+3.
Substitute 2 in for x, which yields 5.