Post on 25-May-2015
POINTS OF DISCONTINUITY (Holes and Asymptotes)
Points of discontinuity occur when the denominator of a function equals zero.
2
2
2
21.
2
32.
4 3
13.
2
x xy
x
xy
x x
xy
x x
1
1 2
xy
x x
3
3 1
xy
x x
2 1
2
x xy
x
Horizontal AsymptotesNote: The graph of a rational function has at most one horizontal asymptote.
If the degree of the numerator is less than the denominator, the horizontal asymptote is y = 0.
If the degree of the numerator is more than the denominator, there is NO horizontal asymptote .
If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is y = a / b , where a and b are the leading coefficients of the numerator and the denominator.
1y
xExample:
2
2
2
2 1
x xy
x x
Example:
2 1y x Example:
EXAMPLE: Finding the Slant/Oblique Asymptote of a Rational Function Find the slant asymptotes of f (x)
2 4 5 .3
x xx
Solution Because the degree of the numerator, 2, is exactly one more than the degree of the denominator, 1, the graph of f has a slant asymptote. To find the equation of the slant asymptote, divide x 3 into x2 4x 5:
2 1 4 51 3 3 1 1 8
32
81 13
3 4 5
xx
x x x
Remainder
moremore
1 2
1 1
x xy
x x
2
2
2
1
x xy
x
Graph the equation.
2
1
2
xy
x x
Graph the equation.
1
1 2
xy
x x