Limits question and finding oblique asymptote

[dustinm]

Question: Guess the oblique asymptote of the graph f(x) for x→∞. Write down the limit you have to compute to prove that your guess is correct.

f(x)= $\sqrt{(x^{4}+1)/(x^{2}-1)}$
so the limit would be: lim x→∞ $\sqrt{(x^{4}+1)/(x^{2}-1)}$

I sketched out a graph but I just have no clue how to compute for the oblique asymptote. The professor explained that you have to use the polynomials in long division but I don't fully understand how to yet.

 

[Mark44]

Question: Guess the oblique asymptote of the graph f(x) for x→∞. Write down the limit you have to compute to prove that your guess is correct.

f(x)= $\sqrt{(x^{4}+1)/(x^{2}-1)}$
so the limit would be: lim x→∞ $\sqrt{(x^{4}+1)/(x^{2}-1)}$

I sketched out a graph but I just have no clue how to compute for the oblique asymptote. The professor explained that you have to use the polynomials in long division but I don't fully understand how to yet.

An oblique asymptote is a straight line y = ax + b that the graph of the function approaches for large x or very negative x.

Use polynomial long division to carry out the division of the rational expression inside the radical. You should get x² + some other terms. Then, factor out x² from each term inside the radical so that you have x²(1 + <other stuff>). At this point you can simplify the radical somewhat.

I'm sure there's a topic on wikipedia for polynomial long division. Open wikipedia and do a search using "polynomial long division" if you don't understand this process.