# Linear oblique asymptote

## Homework Statement

what is the linear oblique asymptote of (x^5+x^3+2)/(x^4-1)
?

## Homework Equations

x-a/p(x) = q(x) +remainder

## The Attempt at a Solution

I put in all the placeholders for the divisor and the numerator and got x as the equation for the linear oblique asymptote?? Is that right??

## Answers and Replies

Mentallic
Homework Helper
We would let

$$\frac{x^5+x^3+2}{x^4-1}\equiv \frac{(ax+b)(x^4-1)+p(x)}{x^4-1} = ax+b + \frac{p(x)}{x^4-1}$$

Where p(x) is a cubic polynomial or less (doesn't matter what it is exactly).

If we expanded (ax+b)(x4-1) then we get

$$ax^5+bx^4-ax-b$$

But we ignore the -ax-b term because that will be a part of p(x) which we've already said we don't care about. So we want the constant a to be chosen such that $ax^5=x^5$ since the coefficient of $x^5$ on the LHS must be equal to the RHS, hence a=1, and b must be chosen such that $bx^4=0$ for the same reason, hence b=0.

But we ignore the -ax-b term because that will be a part of p(x) which we've already said we don't care about.

Is the answer that I got correct? Thanks for taking the time to answer. I know how to do it just wondering if it's correct

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HallsofIvy
Science Advisor
Homework Helper
Yes, y= x is the "liner oblique asymptote".

Mark44
Mentor
Is the answer that I got correct? Thanks for taking the time to answer. I know how to do it just wondering if it's correct
No, this isn't correct. It's the equation of the rational function you started with.

To find the oblique asymptote, either do what Mentallic suggested or carry out the long division to get x + a proper rational function. In a proper rational function, the degree of the numerator is less than that of the denominator.

Yes, y= x is the "liner oblique asymptote".

Ok thank you