when P(x) and Q(x)
are polynomials with the same degrees. We would like to investigate the graph of when P(x) and Q(x)
are polynomials with the same degrees.
Example 1: Investigate the graph of 
(1) First, notice that the vertical asymtote for f is x=-1.
Case 1: As 
say x=-0.999, 
Case 2: As 
say x=-1.001, 
(2) Notice that the horizontal asymtote of f is y=2 (compare the ratio between numerator and denominator.)
Case 1; As 
say x=100,000, 
which means that the graph will approach the
horizontal line y=2 from below.
Case 2; As 
say x=-100,000, 
which means that the graph will approach the
horizontal line y=2 from above. Use the following Maple syntax:
plot((2*x-1)/(x+1),x=-2..2, thickness=3);
to verify your answer.
Example 2: Let 
Graph f.
(1) The vertical asymtotes for f are x=1 and x=2.
Case 1: As 
say x=1.0001, 
Case 2: As 
say x=0.9999, 
Case 3: As 
say x=2.0001, 
Case 4: As 
say x=1.9999, 
(2) The horizontal asymtote for f is y=1.
Case 1: As say x=100,000, 
which means the graph of f
approaches to y=1 from above.
Case 2: As say x=-100,000, 
which means the graph of f approaches
to y=1 from below.
(3) (harder questions): Can you find the x- intercepts and find the intervals where f(x)>0 and f(x)<0.
(4) Use the folloing Maple syntax:
plot((x^2-2)/((x-1)*(x-2)),x=-2..4,y=-10..10, thickness=3);
to check your answer.