Math - Domain and Asymptotes

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.

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  1. 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:
    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:

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  2. x^2+x-6/x+3

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