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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.

 Apr 18, 2020
 #1
avatar+109501 
+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    \(g(x)=-3x^2-9\)

 

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

 

 Apr 18, 2020
 #2
avatar+95 
+1

Thank you so much Melody! 

littlemixfan  Apr 18, 2020
 #3
avatar+109501 
+1

You are welcome :)

Melody  Apr 18, 2020

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