For each rational function below, select the long-run behavior of the numerator and of the denominator. Then, determine the long-run behavior of the rational function.

a) $f(x) = \dfrac{4 x^2-4x+5}{4 x^2+5 x-10}$

Long-run behavior of numerator:
As $x$ goes to $+\infty$, the numerator of $f(x)$ goes to .
As $x$ goes to $-\infty$, the numerator of $f(x)$ goes to .

Long-run behavior of denominator:
As $x$ goes to $+\infty$, the denominator of $f(x)$ goes to .
As $x$ goes to $-\infty$, the denominator of $f(x)$ goes to .

Long-run behavior of rational function:
As $x$ goes to $+\infty$, $f(x)$ goes to .
As $x$ goes to $-\infty$, $f(x)$ goes to .

b) $g(x) = \dfrac{x^3-3x^2}{x^2+5 }$

Long-run behavior of numerator:
As $x$ goes to $+\infty$, the numerator of $g(x)$ goes to .
As $x$ goes to $-\infty$, the numerator of $g(x)$ goes to .

Long-run behavior of denominator:
As $x$ goes to $+\infty$, the denominator of $g(x)$ goes to .
As $x$ goes to $-\infty$, the denominator of $g(x)$ goes to .

Long-run behavior of rational function:
As $x$ goes to $+\infty$, $g(x)$ goes to .
As $x$ goes to $-\infty$, $g(x)$ goes to .

c) $h(x) = \dfrac{4 x}{x^4-4 x^3-1}$

Long-run behavior of numerator:
As $x$ goes to $+\infty$, the numerator of $h(x)$ goes to .
As $x$ goes to $-\infty$, the numerator of $h(x)$ goes to .

Long-run behavior of denominator:
As $x$ goes to $+\infty$, the denominator of $h(x)$ goes to .
As $x$ goes to $-\infty$, the denominator of $h(x)$ goes to .

Long-run behavior of rational function:
As $x$ goes to $+\infty$, $h(x)$ goes to .
As $x$ goes to $-\infty$, $h(x)$ goes to .

Hint: