Question

a function $f(x)$ is said to have a jump discontinuity at $x = a$ if: 1. $lim_{x
ightarrow a^{-}}f(x)$ exists. 2. $lim_{x
ightarrow a^{+}}f(x)$ exists. 3. the left and right limits are not equal. let $f(x)=\begin{cases}6x - 6&\text{if }x < 5\\frac{1}{x + 4}&\text{if }xgeq5end{cases}$ show that $f(x)$ has a jump discontinuity at $x = 5$ by calculating the limits from the left and right at $x = 5$. $lim_{x
ightarrow5^{-}}f(x)=$ $lim_{x
ightarrow5^{+}}f(x)=$ now for fun, try to graph $f(x)$.

Explanation:

Step1: Calculate left - hand limit

For $x\to5^{-}$, we use $f(x)=6x - 6$.
$\lim_{x\to5^{-}}f(x)=\lim_{x\to5^{-}}(6x - 6)$
Substitute $x = 5$ into $6x - 6$: $6\times5-6=24$.

Step2: Calculate right - hand limit

For $x\to5^{+}$, we use $f(x)=\frac{1}{x + 4}$.
$\lim_{x\to5^{+}}f(x)=\lim_{x\to5^{+}}\frac{1}{x + 4}$
Substitute $x = 5$ into $\frac{1}{x + 4}$: $\frac{1}{5+4}=\frac{1}{9}$.

Answer:

$\lim_{x\to5^{-}}f(x)=24$
$\lim_{x\to5^{+}}f(x)=\frac{1}{9}$