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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
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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
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Thank you so much Melody!

littlemixfan  Apr 18, 2020
#3
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You are welcome :)

Melody  Apr 18, 2020