How do I identify vertical, horizontal, and slant asymptotes from an equation?

Question

For example, if I had to match different equations to each asymptote type, how would I go about doing that?

Answer 1

Vertical asymptotes usually occur when there is a zero in the denominator of a fraction, e.g. tan x at x=pi/2.

Horizontal asymptotes only occur when x is approaching ±∞, e.g. e^-x as x -> ∞

What you have called slant asymptotes occur when some terms in an expression become relatively small by comparison to another term. e.g. (2x²+3)/(x+5) as x -> ∞.; another good example is Stirling's formula for n!

Please note that in what you have called a slant asymptote only the relative difference becomes small; the absolute difference increases without bound.

Answer 2

If you have a good understanding of limits and what happens to the value of a fraction as the denominator approaches zero or what happens to the value of a fraction when the denominator approaches positive infinity or negative infinity,then determining the horizontal or vertical asymptotes of an equation becomes quite simple.

Determining slant asymptotes is a little more complex, but not by much.

To explain all of these and for you to acquire understanding on this subject, it would be best to set up a session with a tutor.

I also found this video that should be able to help you, https://www.youtube.com/watch?v=qGCKjuhA4eQ

The video is kinda boring and it doesn't focus much on acquiring a good understanding, but it does answer your question.