How to find out the end behaviour of an asymptote?

Question

How do you find out where the asymptote goes?

Example: y = 2x+3 / x+2

Answer 1

I think the most straightforward way would be to take a limit as x goes to infinity. In this case, using L'hopital's Rule

lim(x->infty) 2x+3/x+2 = lim(x->infty) 2/1 = 2

If you don't know calculus, the three rules generally are:

If the degree of the numerator is less than the degree of the denominator, the asymptote is y=0

If the degree of the numerator is equal to the degree of the denominator, the asymptote is y=k, where k is the division of the leading coefficients (the leading coefficient is the coefficient in front of the term with highest power in both numerator and denominator; in this case, it is 2/1 = 2).

If the degree of the numerator is greater, there is no horizontal asymptote, and the value of the function goes to infinity.

Answer 2

x = -2 is a vertical asymptote, y = 2 is the horizontal. graph approaches y = 2 from above for x < -2, from below for x > -2.

Answer 3

oh my god i totally just learned this in math today, so the vertical asymptote would be x=-2, and then there's something else with the horizontal asymptote but that really confuses me.