given the following rational function
f(x)= x^2+5x-7 / x-4
state domain, find vertical, horizontal and oblique asymptotes if any. I found the domain to be (-inf.,4)U(4, inf.) and vertical asymptote at x=4 am I right? can someone help me with the rest?
what about the oblique asymptote ?
divide out the right side (I used synthetic division)
f(x) = x + 9 + 29/(x-4)
so the oblique asymtote is
y = x+9
see:
http://www.wolframalpha.com/input/?i=%28x%5E2%2B5x-7%29%2F%28x-4%29
Reiny:
Thanks for the websites - they have helped immensely.
Yes, you are correct regarding the domain and the vertical asymptote. To determine the horizontal and oblique asymptotes (if any), we need to analyze the behavior of the function as it approaches positive and negative infinity.
To find the horizontal asymptote, we examine the degrees of the polynomials in the numerator and denominator. In this case, the degree of the numerator (2) is larger than the degree of the denominator (1). Therefore, there is no horizontal asymptote.
To determine if there is an oblique asymptote, we can perform long division or synthetic division to simplify the rational function. Let's perform long division:
```
x + 9
___________________
x - 4 | x^2 + 5x - 7
- (x^2 - 4x)
----------------
9x - 7
```
So, after dividing, we get:
f(x) = x + 9 + (9x - 7) / (x - 4)
Now, as x approaches positive or negative infinity, the term (9x - 7) / (x - 4) becomes negligible compared to the x + 9 term. Therefore, the oblique asymptote is y = x + 9.
To summarize:
- Domain: (-∞, 4) ∪ (4, ∞)
- Vertical asymptote: x = 4
- Horizontal asymptote: None
- Oblique asymptote: y = x + 9