Asymptote: a line that a curve gets closer and closer to.

Polynomial functions don't have any.

Vertical asymptote:

Rational functions
Example:

Logarithmic functions: one-sided vertical asymptote

Variants:

Infinite number of vertical asymptotes, e.g. tan:
  
Also sec and csc

"End behavior":
What happens as x approaches ±∞.
Horizontal asymptote:
on one side only, or the same HA on both sides, or different HA on each side.

2 same horizontal asymptotes:

Rational functions
If a rational function has a H.A., both "ends" approach the same H.A.

Graph can cross horizontal or oblique asymptote but not vertical asymptotes. Find x such that R(x)=horizontal asymptote

Gaussian functions (e^-x2) have H.A. both sides same:

2 different horizontal asymptotes:

   arctan has at ±π/2.
  
   x/√(x²+1) has at ±1:
    
   CDF of standard normal function has at 0 and 1:
  
   Logistic functions
  

1-sided horizontal asymptote:

   Exponential functions (b^x). H.A. at 0:
  

Oblique/slant asymptote:

Rational function: If degree of numerator is one more than degree of denominator:
xⁱ⁺¹/xⁱ
E.g.: x²/x    x³/x²    x⁴/x³    x⁵/x⁴ ...
Only one of horizontal and oblique asymptotes are possible.
Divide N by D to get a linear quotient which is the oblique line, ignore remainder.
N÷D = mx+b + r/D

Other end behaviors

Going off to infinities ±∞:
Both ends:
All Polynomial functions:

cosh and sinh:

Rational functions without horizontal or oblique asymptotes.

One end going to an infinity:

Periodic:

Look at:
sin x²
x sin x
1/x sin x
x sin(1/x)
(sin x²)/x
1/(sin x²)
sin √x