Compare and discuss the long-run behaviors of the functions below. In each blank

Question

Compare and discuss the long-run behaviors of the functions below. In each blank, enter either the constant or the polynomial that the rational function behaves like as x rightarrow plusminus infinity: f(x) = x^3 - 5/x^3 - 8, g(x) = x^4 - 5/x^3 - 8, and h(x) = x^2 - 5/x^3 - 8 f(x) will behave like the function y = as x rightarrow plusminus infinity. g(x) will behave like the function y = as x rightarrow plusminus infinity. h(x) will behave like the function y = as x rightarrow plusminus infinity.

Explanation / Answer

f(x) = (x^3 - 5)/(x^3 - 8)

f(x) = ( 1 - 5/x^3)/(1 - 8/x^3)    ( divide numerator and denomiantor by x^3)

x ---: +/- infinty

f(x) = (1 - 0)/(1 - 0) = 1

g(x) = (x^4 - 5)/(x^3 - 8)

( divide numerator and denomiantor by x^4)

g(x) = (1- 5/x^4)/(x - 8/x^4)

x ---: +/- infinty

g(x) = 1 / infinity = 0

h(x) = (x^2 -5)/(x^3 - 8)

h(x) = (1 - 5/x^2)/(x - 8x^2)     ( divide numerator and denomiator by x^2)

= 1/infinty = 0

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