Find the hole(s), if any, for:

Question

f(x)=-4

      -------

       x-10

 

A -10

B 10 

C -2/5

D does not have hole

E 2/5

F None 

Answer 1

The function is undefined at x=10 since x=10 would cause division by zero (a big no-no - only Chuck Norris can divide by zero and you're not Chuck Norris).  So, at x=10 there is either a hole or a vertical asymptote.  If the limit as x approaches 10 from the left = the limit as x approaches 10 from the right, then there is a hole.

 

The limit of the function as x approaches 10:

 

   from the left is positive infinity    <= Chuck Norris counted to infinity...twice

 

   from the right is negative infinity

 

The limits are unequal so there is a vertical asymptote at x=10.

 

Answer 2

Holes happen when a monomial or binomial factor appears in both the numerator and the denominator.  Since it can cancel out of the fraction, it doesn't effect the graph like a factor that only appears in the denominator.

 

Looking at this fraction, you do not have any repeated factors.  Therefore, there are no holes.  (D)

 

Although technically F makes sense as an answer also.