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