Question

find the limit of f(x) = 9/x - 12 as x approaches ∞ and as x approaches -∞.
lim f(x) = - 12 (type a simplified fraction.)
x→∞
lim f(x) =
x→ -∞ (type a simplified fraction.)

Explanation:

Step1: Recall limit property

The limit of $\frac{c}{x}$ as $x
ightarrow\pm\infty$ where $c$ is a constant is 0. Here $c = 9$.

Step2: Find $\lim_{x

ightarrow\infty}f(x)$
We know that $f(x)=\frac{9}{x}-12$. Using the limit - sum rule $\lim_{x
ightarrow a}(u(x)+v(x))=\lim_{x
ightarrow a}u(x)+\lim_{x
ightarrow a}v(x)$. So $\lim_{x
ightarrow\infty}f(x)=\lim_{x
ightarrow\infty}\frac{9}{x}-\lim_{x
ightarrow\infty}12$. Since $\lim_{x
ightarrow\infty}\frac{9}{x} = 0$ and $\lim_{x
ightarrow\infty}12 = 12$, then $\lim_{x
ightarrow\infty}f(x)=0 - 12=-12$.

Step3: Find $\lim_{x

ightarrow-\infty}f(x)$
Again, using the limit - sum rule $\lim_{x
ightarrow-\infty}f(x)=\lim_{x
ightarrow-\infty}\frac{9}{x}-\lim_{x
ightarrow-\infty}12$. Since $\lim_{x
ightarrow-\infty}\frac{9}{x}=0$ and $\lim_{x
ightarrow-\infty}12 = 12$, then $\lim_{x
ightarrow-\infty}f(x)=0 - 12=-12$.

Answer:

$\lim_{x
ightarrow\infty}f(x)=-12$
$\lim_{x
ightarrow-\infty}f(x)=-12$