1. ## Asymptote

Is it correct to say that $f(x) = \frac{4x^2+4x+1}{x^3-x}$ has an oblique asymptote $f(x) = \frac{4}{x}$ and horizontal asymptotes $x = 0$, $x = 1$ and $x = -1$?

2. The degree of the numerator must be one greater than the degree of the denominator in order to have a slant asymptote. In addition, 4/x is not a straight line. The horizontal asymptotes are all y = 0, because the degree of the denominator is greater than the degree of the numerator.

3. Originally Posted by Ackbeet
The degree of the numerator must be one greater than the degree of the denominator in order to have a slant asymptote.
Can you explain this condition bit more, please?

4. As an example: the function

$\frac{x^{2}-1}{x}=x-\frac{1}{x}$

has a slant asymptote of x, because as x gets very large, the 1/x is very small in comparison to the x, and hence becomes negligible. See here for a more full explanation.

5. Originally Posted by Ackbeet
As an example: the function

$\frac{x^{2}-1}{x}=x-\frac{1}{x}$

has a slant asymptote of x, because as x gets very large, the 1/x is very small in comparison to the x, and hence becomes negligible. See here for a more full explanation.
okay, thank you. just one more question, so that I get this.

what can we say about the asymptotes of $f(x) = \frac{x^4+1}{x^3-1}$? I see that it has vertical asymptote at x = 1, x = -1 and x = 0, but I'm not sure what else

6. Why does it have a vertical asymptote at x = -1 and x = 0?

7. Originally Posted by pickslides
Why does it have a vertical asymptote at x = -1 and x = 0?
Sorry, I meant

$f(x) = \frac{x^4+1}{x^3-x}$

8. Originally Posted by Resilient
...

what can we say about the asymptotes of $f(x) = \frac{x^4+1}{x^3-x}$? I see that it has vertical asymptote at x = 1, x = -1 and x = 0, but I'm not sure what else
Originally Posted by Resilient
Sorry, I meant

$f(x) = \frac{x^4+1}{x^3-x}$
1. As you stated correctly the vertical asymptotes correspond to the zeros of the denominator.

2. To get the horizontal or slanted asymptotes transform the term of the function into a complete polynomial and a fraction:
Code:
                                       x^2 + 1
(x^4        +  1) ÷ (x^3 - x) = x + -----------
-(x^4 - x^2)                           x^3 - x
------------
x^2 + 1
The none-fractional part of the term is the term of the slanted asymptote. That means in your case the slanted asymptote has the equation:

$y = x$