Question

find the limit of f(x) = \frac{12}{x}-3 as x approaches \infty and as x approaches -\infty.
lim f(x) =
(typ e a simplified fraction.)

Explanation:

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

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

Step2: Split the limit

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

Answer:

$-3$