Oblique Asymptotes Help!!!

[stxstudent]

I am having trouble figuring out how to solve for oblique asymptotes and I could really use some help. BIG test coming up soon!

There are two problems:

The first is:

f(x)=2xcubed-2xsquared+x+1
_____________________
x-2

and the second:

f(x)= xsquared-2x+1
____________
x

I would really really appreciate a step by step answer on how to solve both or either of these questions.

Thank you in advance to anyone that offers help.

 

[soroban]

Hello, stxstudent!

The key word is Division.

Find the oblique asymptotes.

\(\displaystyle 1)\;\L f(x)\;=\;\frac{2x^3\,-\,2x^2\,+\,x\,+\,1}{x\,-\,2}\)

Use long or synthetic division

\(\displaystyle \;\;\)and we get: \(\displaystyle \L\,f(x)\;=\;2x^2\,+\,2x\,+\,5\,+\,\frac{11}{x\,-\,2}\)


So, as \(\displaystyle x\to\infty\), the fraction approaches 0.

Therefore, the graph approaches the parabola: \(\displaystyle \L\,f(x)\:=\:2x^2\,+\,2x\,+\,5\)

\(\displaystyle 2)\;f(x)\;=\;\frac{x^2\,-\,2x\,+\,1}{x}\)

We can use "short" division on this one . . .

\(\displaystyle \L\;\;f(x)\;=\;\frac{x^2}{x}\,-\,\frac{2x}{x}\,+\,\frac{1}{x}\;=\;x\,-\,2\,+\,\frac{1}{x}\)


As \(\displaystyle x\to\infty,\;\frac{1}{x}\to0\)

Therefore, the graph approaches the line: \(\displaystyle \L\,f(x)\:=\:x\,-\,2\)