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A rational function is a ratio of two polynomials
f(x) = .
Oblique asymptote is a straight line of the form y = ax + b.
Oblique asymptote arises when the degree of the polynomial p(x) in the numerator is 1 unit greater
than the degree of the polynomial q(x) in the denominator.
To get the equation of the oblique asymptote under appropriate condition, you divide the polynomial in the numerator
by the polynomial in the denominator (long division).
Then you get a linear binomial function as a quotient, and it is the desired equation of the oblique asymptote.
Example:
f(x) =
The numerator is p(x) = .
The denominator is q(x) = .
= x - 1 - .
The quotient is x-1, and the equation of the oblique asymptote is y = x-1 in this case.
See the plot below.
Plot y = (the given rational function, red) and y = x-1 (oblique asymptote, green)
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Comment from student: What if the p(x) is degree 8 & q( x) is of degree 3
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My response :
Then there is NO oblique asymptote.
There is a "curvilinear asymptote", instead.
See this Wikipedia article
https://en.wikipedia.org/wiki/Asymptote#Oblique_asymptotes
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It is good that you ask questions. I like it . . .
Do not forget to post your "THANKS" for my teaching.
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I got next your "comment" post with several other questions.
The answers to all these questions are in that Wikipedia article I referred to you.
You will find answers to all your questions in that article.
I don't want to re-tell it here, in my post.
Wikipedia makes it better and in more authoritative way . . .
Happy learning (!)