4. $k(x)=\frac{2x(x - 3)}{(x + 2)^2(x - 1)}$\nleft: $lim_{x\rightarrow-infty}k(x)=0$\nright…

Explanation:

Step1: Analyze degree of numerator and denominator

The numerator of $k(x)=\frac{2x(x - 3)}{(x + 2)^2(x - 1)}=\frac{2x^2-6x}{x^3+3x^2 - 4}$ is a polynomial of degree 2 ($2x^2$ is the leading - term) and the denominator is a polynomial of degree 3 ($x^3$ is the leading - term).

Step2: Use limit rule for rational functions

For a rational function $\frac{f(x)}{g(x)}$ where $\text{deg}(f(x))<\text{deg}(g(x))$, we have $\lim_{x\rightarrow\pm\infty}\frac{f(x)}{g(x)} = 0$. Here, as $x\rightarrow-\infty$ (left - hand limit) and $x\rightarrow\infty$ (right - hand limit), since the degree of the numerator is less than the degree of the denominator, $\lim_{x\rightarrow-\infty}k(x)=0$ and $\lim_{x\rightarrow\infty}k(x)=0$.

Answer:

The left - hand limit $\lim_{x\rightarrow-\infty}k(x)=0$ and the right - hand limit $\lim_{x\rightarrow\infty}k(x)=0$