Question #5018

write a rational function that would have a horizontal asmptote of 5

Expert's answer

Horizontal asymptotes are horizontal lines that the graph of the function approaches as x → ±∞. The horizontal line y = c is a horizontal asymptote of the function y = ƒ(x) if

<img style="width: 211px; height: 27px;" src="data:image/png;base64,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" alt="">

Hence we can pick for example a function f(x)=(5x+1)/(x-4)

It's rational and tends to 5 when x → ±∞

<img style="width: 211px; height: 27px;" src="data:image/png;base64,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" alt="">

Hence we can pick for example a function f(x)=(5x+1)/(x-4)

It's rational and tends to 5 when x → ±∞

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