Holes Vertical Asymptotes and Roots

Where is the hole? Hint: Factor the numerator and denominator expressions.

The holes of a rational function occur when :

• My pencil pokes a hole in the paper, which I am working on.

• There is something left in the denominator.

• The value of f(0).

• A factor in the denominator cancels with a factor in the numerator.

What is the Vertical Asymptotes? Hint: Factor the numerator and denominator expressions.

Find the vertical asymptotes and holes of the function.

• Holes: None;

  VA: x = 1, -3

• Holes: x = -3;

  VA: x = 1

• Holes: x = -3, 1

  VA: None

• Holes: x = 1

  VA: x = -3

Find the vertical asymptote(s), hole(s) and horizontal asymptote.

• V.A.: x = -1
  Holes: x = 2
  H.A.: None

• V.A.: x = 2
  Holes: x = 1
  H.A.: None

• V.A.: x = -1
  Holes: x = 2
  H.A.: y = 1/4

• V.A.: x = 2, -1
  Holes: x = None
  H.A.: None

A vertical asymptote is found by setting the _________= to zero and solving for x  (a)  

• Numerator

• Denominator

• Right side 

• Left side

• x = 2

• x = 7

• x = 2 and x = 7

• x = 0

• Does Not Exist

When identifying a vertical asymptote or hole, what variable do we use?

• x

• y

• none

• the same as the problem

What is the Vertical Asymptotes? 

• x= -5

• x= 5

• x= 2

• x= -2