SOLUTION: What is the horizontal or oblique asymptotes, the vertical asymptotes, describe their meanings, for the following rational function: f(x) = (x^2-5)/(x+3)

f(x) = 

The vertical asymptote is found by setting the denominator = 0

x+3 = 0
  x = -3

That is the equation of a vertical line through -3 on the x axis.

So we draw that, in green:



Since the degree of the numerator, which is 2, is greater 
than the degree of the denominator, which is 1, there is 
no horizontal asymptote.

Since the degree of the numerator, which is 2, is exactly 1
more than the degree of the denominator, which is 1, there 
is an oblique asymptote.

The equation of the oblique asymptote is found by dividing 
the denominator into the numerator by long division, 
ignoring the remainder, and setting y = the quotient:

        x-3
x+3)x²+0x-5
    x²+3x
      -3x-5
      -3x-9
          4

So the oblique asymptote has the equation y = x-3.  
It has intercepts (3,0) and (0,-3) So we draw that 
line (also in green):



Now we get some points (-7,-11), (-5,-10), (-4,-11), (-2,-1),
(-1,-2),(0,), (1,-1), (3,), (5,)

  

Then draw in the graph through those points approacking the
asymptotes:



Edwin