Question

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

Explanation:

Step1: Identify discontinuity point

The graph has a discontinuity at $x=-1$.

Step2: Check if $f(a)$ is defined

At $x=-1$, there is a filled dot at $y=1$, so $f(-1)=1$ (defined).

Step3: Check if $\lim_{x \to a}f(x)$ exists

As $x$ approaches $-1$ from left and right, the graph approaches the open dot at $y=-3$. So $\lim_{x \to -1}f(x)=-3$ (exists).

Step4: Check if $\lim_{x \to a}f(x)=f(a)$

We have $\lim_{x \to -1}f(x)=-3$ and $f(-1)=1$. Since $-3
eq 1$, this equality fails.

Answer:

$\boldsymbol{\lim_{x \to a} f(x) = f(a)}$