Asymptotes
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Key Questions

An asymptote is a value of a function that you can get very near to, but you can never reach.
Let's take the function
#y=1/x#
graph{1/x [10, 10, 5, 5]}
You will see, that the larger we make#x# the closer#y# will be to#0#
but it will never be#0# #(x>oo)# In this case we call the line
#y=0# (the xaxis) an asymptoteOn the other hand,
#x# cannot be#0# (you can't divide by#0# )So the line
#x=0# (the yaxis) is another asymptote. 
To Find Vertical Asymptotes:
In order to find the vertical asymptotes of a rational function, you need to have the function in factored form. You also will need to find the zeros of the function. For example, the factored function
#y = (x+2)/((x+3)(x4)) # has zeros at x =  2, x =  3 and x = 4.*If the numerator and denominator have no common zeros, then the graph has a vertical asymptote at each zero of the denominator. In the example above
#y = (x+2)/((x+3)(x4)) # , the numerator and denominator do not have common zeros so the graph has vertical asymptotes at x =  3 and x = 4.*If the numerator and denominator have a common zero, then there is a hole in the graph or a vertical asymptote at that common zero.
Examples:
1.#y= ((x+2)(x4))/(x+2)# is the same graph as y = x  4, except it has a hole at x =  2.2.
#y= ((x+2)(x4))/((x+2)(x+2)(x4))# is the same as the graph of#y = 1/(x + 2),# except it has a hole at x = 4. The vertical asymptote is x =  2.To Find Horizontal Asymptotes:

The graph has a horizontal asymptote at y = 0 if the degree of the denominator is greater than the degree of the numerator. Example: In
#y=(x+1)/(x^2x12)# (also#y=(x+1)/((x+3)(x4))# ) the numerator has a degree of 1, denominator has a degree of 2. Since the degree of the denominator is greater, the horizontal asymptote is at#y=0# . 
If the degree of the numerator and the denominator are equal, then the graph has a horizontal asymptote at
#y = a/b# , where a is the coefficient of the term of highest degree in the numerator and b is the coefficient of the term of highest degree in the denominator. Example: In#y=(3x+3)/(x2)# the degree of both numerator and denominator are both 1, a = 3 and b = 1 and therefore the horizontal asymptote is#y=3/1# which is#y = 3# 
If the degree of the numerator is greater than the degree of the denominator, then the graph has no horizontal asymptote.


Example 1:
#f(x)=x^2/{(x+2)(x3)}# Vertical Asymptotes:
#x=2# and#x=3#
Horizontal Asymptote:#y=1#
Slant Asymptote: NoneExample 2:
#g(x)=e^x# Vertical Asymptote: None
Horizontal Asymptote:#y=0#
Slant Asymptote: NoneExample 3:
#h(x)=x+1/x# Vertical Asymptote:
#x=0#
Horizontal Asymptote: None
Slant Asymptote:#y=x# I hope that this was helpful.