Question

find the limit of f(x) = $\frac{5}{x}$ - 2 as x approaches $infty$ and as x approaches - $infty$.
lim f(x) = - 2
x→∞
(type a simplified fraction.)
lim f(x) =
x→ - ∞
(type a simplified fraction.)

Explanation:

Step1: Recall limit rule for $\frac{1}{x}$

As $x\to\pm\infty$, $\lim_{x\to\pm\infty}\frac{1}{x}=0$.

Step2: Analyze $\lim_{x\to-\infty}f(x)$

We have $f(x)=\frac{5}{x}-2$. Using the limit - sum rule $\lim_{x\to a}(u(x)+v(x))=\lim_{x\to a}u(x)+\lim_{x\to a}v(x)$, where $u(x)=\frac{5}{x}$ and $v(x)= - 2$. Then $\lim_{x\to-\infty}f(x)=\lim_{x\to-\infty}\frac{5}{x}+\lim_{x\to-\infty}(-2)$. Since $\lim_{x\to-\infty}\frac{5}{x}=5\lim_{x\to-\infty}\frac{1}{x}=0$ and $\lim_{x\to-\infty}(-2)=-2$, we get $\lim_{x\to-\infty}f(x)=0 - 2=-2$.

Answer:

-2