Some Help Please!

D

Danielle058

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Aug 19, 2005
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Alright the problem that I've been trying to solve is, ((x^3)-2)/(x-1)

And I'm supposed to find the holes; if any exist.. I've determined that there are holes because the greater power, x^3 is on top and not in the denominator. Therefore, there has to be holes, I just don't know how to find them... so if anyone can help me with this I'd appreciate it!

Thanks! 8-)
 
Alright the problem that I've been trying to solve is, ((x^3)-2)/(x-1)

And I'm supposed to find the holes; if any exist..
Click to expand...

I'm sure you mean

y = ((x^3)-2)/(x-1)


Holes exist when the numerator and denominator have common factors containing x terms. Vertical asymptotes exist due to constraints on the denominator (the denominator cannot equal zero). In this problem, x-1 is not a factor of x^3-2. Therefore, you do not have a hole; you have a vertical asymptote at x = 1.
 
hm, I thought that whenever the power is greater on the top than on the bottom there is not horizontal asymptote rather, there are holes and you are looking for the holes because there are no horizontal asymptotes and the exponents are higher in the numberator than denominator? :?
 
...you are looking for the holes because there are no horizontal asymptotes...
Click to expand...

No -- horizontal asymptotes have nothing to do with holes.

Having x-1 in the denominator means there must be either a hole or a VERTICAL asymptote at x =1.
 
Horizontal Asymptotes: Numerator Degree <= Denominator Degree

Other Kinds of Asymptotes: Numerator Degree > Denominator Degree

Vertical Asymptotes: Denominator is zero

UNLESS

Holes: Same factor (and same degree of factor) in Numerator and Denominator

f(x) = (x^3 + 2)/(x-1)

x = 1 makes the denominator zero (0). There is no such result in the numerator. x = 1 is a vertical asymptote. There are no holes.

Numerator Degree = Denominator Degree + 2. This makes for a parabolic (curved) asymptote. Long division shows f(x) = x^2 + x + 1 + 3/(x-1), making the asymptote, g(x) = x^2 + x + 1
 
(x^2 -1)/(x^3- 2x^2 + x)

Do any of them have holes?
Click to expand...

You must factor both the numerator and denominator and see if they have any common factors with x in them. If they do, you have a hole.
 
Read my description of "Holes" above. It is worded with some care.

Though Numerator and Denominator share a factor of (x-1), they do not share the same number of factors of (x-1). There is NO hole in this one. x = 1 is a Vertical Asymptote.
 
Though Numerator and Denominator share a factor of (x-1), they do not share the same number of factors of (x-1). There is NO hole in this one. x = 1 is a Vertical Asymptote.
Click to expand...

Thank you, TK, for correcting my hasty response. :oops: :D

Danielle, please note that x=0 is also a vertical asymptote.
 
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