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**Topics:**Rational function, Function, Limit of a function

**Pages:**33 (573 words)

**Published:**October 13, 2014

Asymptotes, and Holes

Dr. Litan Kumar Saha

Assistant Professor

Department of Mathematics

University of Dhaka

Rational Expression

It is the quotient of two polynomials.

A rational function is a function defined by a

rational expression.

Examples:

y

x2

x5

y

3x2 2 x 5

x3 4 x2 5 x 7

Not Rational:

y

4x

x2

y

x

x2 5

Find the domain:

Graph it:

f ( x)

1

x

Find the domain:

Graph it:

f ( x)

1

x

Find the domain:

Graph it:

f ( x)

1

x2

Find the domain:

Graph it:

f ( x)

1

x2

Vertical Asymptote

If x – a is a factor of the denominator of a

rational function but not a factor of the

numerator, then x = a is a vertical asymptote

of the graph of the function.

Find the domain:

Graph it using the

graphing calculator.

What do you see?

f ( x)

x3

x x 12

2

Find the domain:

Graph it using the

graphing calculator.

What do you see?

f ( x)

x3

x x 12

2

Hole (in the graph)

If x – b is a factor of both the numerator and

denominator of a rational function, then there

is a hole in the graph of the function where

x = b, unless x = b is a vertical asymptote.

The exact point of the hole can be found by

plugging b into the function after it has been

simplified.

Find the domain and identify

vertical asymptotes & holes.

f ( x)

x 1

x2 2x 3

Find the domain and identify

vertical asymptotes & holes.

f ( x)

x 1

x2 2x 3

Find the domain and identify

vertical asymptotes & holes.

f ( x)

x

x2 4

Find the domain and identify

vertical asymptotes & holes.

f ( x)

x

x2 4

Find the domain and identify

vertical asymptotes & holes.

f ( x)

x5

2x2 x 3

Find the domain and identify

vertical asymptotes & holes.

f ( x)

x5

2x2 x 3

Find the domain and identify

vertical asymptotes & holes.

f ( x)

3 2x x2

x2 x 2

Find the domain and identify

vertical asymptotes & holes.

f ( x)

3 2x x2

x2 x 2

Horizontal Asymptotes

Degree of numerator = Degree of denominator

Horizontal Asymptote:

coefficient of denominator

Degree of numerator < Degree of denominator

Horizontal Asymptote:

y

coefficient of numerator

y0

Degree of numerator > Degree of denominator

Horizontal Asymptote:

none

Find all asymptotes & holes & then graph:

f ( x)

x2

2x 3

Find all asymptotes & holes & then graph:

f ( x)

x2

2x 3

Find all asymptotes & holes & then graph:

f ( x)

x 1

x 2x 3

2

Find all asymptotes & holes & then graph:

f ( x)

x 1

x 2x 3

2

Find all asymptotes & holes & then graph:

f ( x)

x2 3x 4

x4

Find all asymptotes & holes & then graph:

f ( x)

x2 3x 4

x4

Find all asymptotes & holes & then graph:

f ( x)

3x2 7 x 6

x2 9

Find all asymptotes & holes & then graph:

f ( x)

3x2 7 x 6

x2 9

Find all asymptotes & holes & then graph:

f ( x)

x 3

x2

2

Find all asymptotes & holes & then graph:

f ( x)

x 3

x2

2

Find all asymptotes & holes & then graph:

f ( x)

3x2

2x2 5

Find all asymptotes & holes & then graph:

f ( x)

3x2

2x2 5

Find all asymptotes & holes & then graph:

f ( x)

x

x2 2x 3

Find all asymptotes & holes & then graph:

f ( x)

x

x 2x 3

2

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