Question

what are the left and right end behaviors of the rational function (y values)? 4. $k(x)=\frac{2x(x - 3)}{(x + 2)^2(x - 1)}$ left: __________ right: __________

Explanation:

Step1: Expand the numerator and denominator

The numerator $2x(x - 3)=2x^{2}-6x$. The denominator $(x + 2)^{2}(x - 1)=(x^{2}+4x + 4)(x - 1)=x^{3}+4x^{2}+4x-x^{2}-4x - 4=x^{3}+3x^{2}-4$.

Step2: Analyze the degrees of the numerator and denominator

The degree of the numerator $n = 2$ and the degree of the denominator $m=3$. When $m>n$, as $x\to\pm\infty$, $y\to0$.

Step3: Determine left - hand and right - hand behavior

As $x\to-\infty$, we consider the sign of the function. The leading - term of the numerator is $2x^{2}$ (positive for large $|x|$) and the leading - term of the denominator is $x^{3}$ (negative for $x\to-\infty$), so $y\to0^{-}$.
As $x\to+\infty$, the leading - term of the numerator is $2x^{2}$ (positive) and the leading - term of the denominator is $x^{3}$ (positive), so $y\to0^{+}$.

Answer:

Left: $y\to0^{-}$
Right: $y\to0^{+}$