# Thread: Finding Verticle asymptotes, holes and, and horizontal asymptotes

1. ## Finding Verticle asymptotes, holes and, and horizontal asymptotes

im having a really difficult time finding out how to find the verticle asymptote and holes and horizontal asymptote. its really frustrating

for exmaple here are some questions:

2. ## Re: Finding Verticle asymptotes, holes and, and horizontal asymptotes

$y = \frac{(x-1)(x+2)(x+3)}{2(x+1)(x+2)(x-3)}$

where are the vertical asymptotes?

what is the horizontal asymptote?

where is the point discontinuity (hole) ?

3. ## Re: Finding Verticle asymptotes, holes and, and horizontal asymptotes

out of my little understanding of it (my instructor sucks)
i can say that
horizontal asymptote at 1/2
vertical asymptotes at -1,-2, and 3
holes at........i dont even know. LOL

4. ## Re: Finding Verticle asymptotes, holes and, and horizontal asymptotes

Originally Posted by Recyclop
out of my little understanding of it (my instructor sucks)
i can say that
horizontal asymptote at 1/2
vertical asymptotes at -1,-2, and 3
holes at........i dont even know. LOL
horizontal asymptote at y = 1/2 ... it's the equation of a line

vertical asymptotes at x = -1 and x = 3 , "hole" at x = -2 ... note the common factor in the numerator and denominator.

OK ... using one of your problems

$g(x) = \frac{6x^2}{x^4 - 4x^2} = \frac{6x^2}{x^2(x^2 - 4)} = \frac{6x^2}{x^2(x-2)(x+2)}$

horizontal asymptote at y = ?

two vertical asymptotes ... where?

one "hole" ... where?

5. ## Re: Finding Verticle asymptotes, holes and, and horizontal asymptotes

so horizontal asymptote @ y=6
verticle asymptotes at x=+2 and x=-2
the hole would be at.....X^2 would be the only common factor i see. but that would be sqrt0.... so there is no hole?

6. ## Re: Finding Verticle asymptotes, holes and, and horizontal asymptotes

horizontal asymptote is y = 0 ... the degree of the numerator < degree of the denominator

vertical asymptotes correct

hole at x = 0

7. ## Re: Finding Verticle asymptotes, holes and, and horizontal asymptotes

so after class today i got a good understanding of asymptotes and what not.
only thing that still doesnt make sense is holes. i understand that a hole occurs when you have common factors but what do i do to find the point and what does a hole mean?

8. ## Re: Finding Verticle asymptotes, holes and, and horizontal asymptotes

A "hole" is really a point discontinuity ... the discontinuity occurs at the x-value that makes the common factor in the numerator and denominator equal to zero.

for example, the function $y = \frac{x^2-1}{x-1} = \frac{(x+1)(x-1)}{x-1}$ has a "hole" at $x = 1$. the graph of the function will look like the line $y = x+1$ except for the "hole" at $x = 1$ ...

note the graph made on my TI calculator ... see the "hole" ?

9. ## Re: Finding Verticle asymptotes, holes and, and horizontal asymptotes

ya i see, so how do we determine the y value? is it just whatever number happens to be there if we plug in 1 for the complete function?

10. ## Re: Finding Verticle asymptotes, holes and, and horizontal asymptotes

sub in x = 1 after dividing out the common factor ... in this case, y = x+1 = 1 + 1 = 2 ... the "hole" is at (1,2).

11. ## Re: Finding Verticle asymptotes, holes and, and horizontal asymptotes

so if there are holes, are the common factors canceled first? then there would be no Vertical asymptotes?

12. ## Re: Finding Verticle asymptotes, holes and, and horizontal asymptotes

the function $y = \frac{(x+1)(x-2)}{(x-3)(x+1)}$ has both a hole at one x-value and a vertical asymptote at another x-value ... where?

13. ## Re: Finding Verticle asymptotes, holes and, and horizontal asymptotes

hole at 1/2. but would 3 and -1 still be verticle asymptotes and would -1 still be a root?

14. ## Re: Finding Verticle asymptotes, holes and, and horizontal asymptotes

sorry, but no ...

there is a hole at x = -1 (the value of x that makes the factor common to the numerator and denominator 0), the y-value of the hole is 3/4

there is one vertical asymptote at x = 3

15. ## Re: Finding Verticle asymptotes, holes and, and horizontal asymptotes

ok i see now, thank you very much!