Let \(f(x) = 3\cdot\frac{x^4+x^3+x^2+1}{x^2+x-2}. \)

Give a polynomial g(x) so that f(x)+g(x) has a horizontal asymptote of 0 as x approaches positive infinity.

littlemixfan Apr 18, 2020

#1**+3 **

I mucked around with this quite a bit before I came up with a solution.

First I did the division and found \(f(x)=3(x^2+3)+\frac{3(-3x-5)}{x^2+x-2}\)

If I add \(-3(x^2+3)\)

then the resulting expression will be \( \frac{3(-3x-5)}{x^2+x-2}\)

Now with this latest expression the degree of the numerator is 1 and the degree of the denominator is 2

As x tends to infinity both the numerator and the denominator tend to infinity also but the bottom is approaching infinity at a much faster rate (since the degree is higher.) so the expression will tend to 0 as x tends to infinity.

Hence one solution is \(g(x)=-3x^2-9\)

\(\displaystyle \lim_{x\rightarrow\infty}\;f(x)+g(x)=0^-\)

Melody Apr 18, 2020