find the limit of f(x) = \\frac{12}{x}-3 as x approaches \\infty and as x approaches -\\infty. \nlim f(x) =…

Explanation:

Step1: Recall limit rule for $\frac{c}{x}$

As $x\rightarrow\pm\infty$, $\lim_{x\rightarrow\pm\infty}\frac{c}{x}=0$ where $c$ is a constant. Here $c = 12$.

Step2: Split the limit

We know that $\lim_{x\rightarrow\infty}f(x)=\lim_{x\rightarrow\infty}(\frac{12}{x}-3)=\lim_{x\rightarrow\infty}\frac{12}{x}-\lim_{x\rightarrow\infty}3$. Since $\lim_{x\rightarrow\infty}\frac{12}{x} = 0$ and $\lim_{x\rightarrow\infty}3=3$, we have $\lim_{x\rightarrow\infty}(\frac{12}{x}-3)=0 - 3=-3$.

Answer:

$-3$