Question

question 8
the graph below is the function $f(x)$
determine which one of the following rules for continuity is first violated at $a = -1$.
$\bigcirc$ $f(a)$ is defined.
$\bigcirc$ $lim_{x \to a} f(x)$ exists.
$\bigcirc$ $lim_{x \to a} f(x) = f(a)$.

Explanation:

Step1: Check if $f(a)$ is defined

At $a=-1$, the graph has a filled dot at $y=3$, so $f(-1)=3$, meaning $f(a)$ is defined.

Step2: Check if $\lim_{x \to a} f(x)$ exists

As $x$ approaches $-1$ from the left, the function approaches $1$; as $x$ approaches $-1$ from the right, the function approaches $3$. Since $\lim_{x \to -1^-} f(x)
eq \lim_{x \to -1^+} f(x)$, $\lim_{x \to -1} f(x)$ does not exist.

Step3: Identify first violated rule

The first rule ($f(a)$ is defined) is satisfied, so the first violated rule is the existence of the limit.

Answer:

$\boldsymbol{\lim_{x \to a} f(x) \text{ exists.}}$