Posted by **Greg - PV** on Thursday, June 9, 2011 at 7:44pm.

Given the following rational function:

f(x) = (x^2 + 6x - 8) / (x – 5)

(a) state the domain.

(b) find the vertical and horizontal asymptotes, if any.

(c) find the oblique asymptotes, if any.

- Math - Domain and Asymptotes -
**MathMate**, Friday, June 10, 2011 at 9:10am
Given the rational function:

f(x) = (x^2 + 6x - 8) / (x – 5)

a. Domain

The domain of a rational function is all real numbers *minus* points where the denominator (x-5) become zero. Here the point to be removed is x-5=0, or x=5.

The answer in interval notation would be:

(-&infin,5)∪(5,∞)

which is essentially all real less x=5.

b. Asymptotes

Vertical asymptotes occur when the denominator becomes zero. There is one such asymptote for f(x).

Hint: this point has been identified in part (a) above.

Horizontal asymptotes are limits of f(x) as x→-∞ or x→∞.

If these limits do not exist, there are no horizontal asymptotes.

Hint: Evaluate Lim x→±∞ and see if the limits exist.

3. oblique asymptotes

Oblique asymptotes exist when the leading term of the numerator divided by the leading term of the denominator yields a linear term (i.e. of the form kx), where k≠0.

Here, the leading term of the numerator is x^2, and that of the denominator is x.

The quotient is therefore x^2/x=x (k=1).

The oblique asymptote is therefore y=x.

Follow link below for the graph:

http://imageshack.us/photo/my-images/221/1307663049.png/

- Math - Domain and Asymptotes -
**Anonymous**, Thursday, May 16, 2013 at 9:10am
x^2+x-6/x+3

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