Question

the graph below is the function $f(x)$
determine which one of the following rules for continuity is violated first at $x = 2$.
\\(\bigcirc f(a)\\) is defined.
\\(\bigcirc \lim\limits_{x \to a} f(x)\\) exists.
\\(\bigcirc \lim\limits_{x \to a} f(x) = f(a)\\).
question help: \\(\boxed{\text{video}}\\)

Explanation:

Step1: Recall Continuity Rules

A function \( f(x) \) is continuous at \( x = a \) if three conditions are met:
1. \( f(a) \) is defined.
2. \( \lim_{x \to a} f(x) \) exists.
3. \( \lim_{x \to a} f(x) = f(a) \).

Step2: Analyze the Graph at \( x = 2 \)

- For \( f(a) \) is defined: At \( x = 2 \), there is an open circle, which means \( f(2) \) is not defined (open circle indicates a point is not included, so the function has no value there).
- For \( \lim_{x \to 2} f(x) \) exists: To check the limit, we look at the left - hand limit (as \( x \) approaches 2 from the left) and the right - hand limit (as \( x \) approaches 2 from the right). From the graph, as \( x \) approaches 2 from both the left and the right, the function approaches the same \( y \) - value (the \( y \) - value near the open circle). So the limit \( \lim_{x \to 2} f(x) \) exists.
- For \( \lim_{x \to 2} f(x)=f(2) \): Since \( f(2) \) is not defined (open circle), we can't even start to check this condition until \( f(2) \) is defined. But the first condition that fails here is the definition of \( f(a) \) at \( a = 2 \)? Wait, no. Wait, the options are about which rule is violated first. Wait, the first condition for continuity is \( f(a) \) is defined. But let's re - examine. Wait, the open circle at \( x = 2 \) means \( f(2) \) is not defined. But wait, the three conditions: the first condition is \( f(a) \) is defined. But let's check the options again. Wait, the options are:
- \( f(a) \) is defined. (Is this condition violated? Yes, because at \( x = 2 \), \( f(2) \) is not defined (open circle))
- \( \lim_{x \to a} f(x) \) exists. (As we saw, the left and right limits exist and are equal, so this condition is satisfied)
- \( \lim_{x \to a} f(x)=f(a) \). (We can't check this because \( f(a) \) is not defined)

But wait, the question is which rule is violated first. The first rule for continuity is that \( f(a) \) must be defined. Since at \( x = 2 \), \( f(2) \) is not defined (open circle), the rule \( f(a) \) is defined is violated. Wait, no, wait the options are presented as the three conditions. Wait, the first option is " \( f(a) \) is defined" - is this condition met? No, because at \( x = 2 \), there's an open circle, so \( f(2) \) is not defined. The second option: \( \lim_{x \to 2} f(x) \) exists? Let's see, as \( x \) approaches 2 from the left, the function comes from the left - hand side of \( x = 2 \) towards the \( y \) - value of the open circle, and as \( x \) approaches 2 from the right, it also comes towards the same \( y \) - value. So the left - hand limit and right - hand limit are equal, so the limit exists. The third option: \( \lim_{x \to 2} f(x)=f(2) \). But \( f(2) \) is not defined, so we can't have this equality. But the first condition ( \( f(a) \) is defined) is violated because \( f(2) \) is not defined. Wait, but the way the options are phrased: the first option is " \( f(a) \) is defined" - is this a rule that is violated? Yes, because \( f(2) \) is not defined. Wait, no, the three conditions for continuity are:

1. \( f(a) \) is defined.

2. \( \lim_{x ightarrow a}f(x) \) exists.

3. \( \lim_{x ightarrow a}f(x)=f(a) \)

At \( x = 2 \), the first condition ( \( f(2) \) is defined) is violated (open circle means no definition), the second condition (limit exists) is satisfied (left and right limits are equal), and the third condition can't be satisfied because the first condition is not satisfied. So the rule that is violated first is \( f(a) \) is defined? Wait, no, wait the options are presented as the three conditions, and we have to see which one is violated. Wait, the first option is " \( f(a) \) is defined" - is this condition met? No, so this rule is violated. The second option: \( \lim_{x ightarrow a}f(x) \) exists - this is met. The third option: \( \lim_{x ightarrow a}f(x)=f(a) \) - since \( f(a) \) is not defined, we can't check this, but the first violation is the non - definition of \( f(a) \). Wait, but wait, maybe I made a mistake. Wait, the open circle at \( x = 2 \): the function has no value at \( x = 2 \), so \( f(2) \) is not defined. So the first condition for continuity ( \( f(a) \) is defined) is violated.

Wait, but let's re - read the question: "Determine which one of the following rules for continuity is violated first at \( x = 2 \)."

The three rules for continuity are:

1. \( f(a) \) is defined.

2. \( \lim_{x ightarrow a}f(x) \) exists.

3. \( \lim_{x ightarrow a}f(x)=f(a) \)

At \( x = 2 \):

- Rule 1: \( f(2) \) is not defined (open circle), so this rule is violated.

- Rule 2: The left - hand limit (as \( x \) approaches 2 from the left) and the right - hand limit (as \( x \) approaches 2 from the right) are equal (the function approaches the same \( y \) - value from both sides), so the limit exists. So this rule is not violated.

- Rule 3: Since \( f(2) \) is not defined, we can't have \( \lim_{x ightarrow 2}f(x)=f(2) \), but this rule can't be violated until \( f(2) \) is defined. The first violation is the non - definition of \( f(2) \), so the rule \( f(a) \) is defined is violated.

Wait, no, wait the options are:

- \( f(a) \) is defined. (Is this a rule that is violated? Yes, because \( f(2) \) is not defined)

- \( \lim_{x ightarrow a}f(x) \) exists. (Not violated)

- \( \lim_{x ightarrow a}f(x)=f(a) \) (Can't be violated yet because \( f(a) \) is not defined)

So the rule that is violated first is \( f(a) \) is defined. Wait, but the option is " \( f(a) \) is defined" - so the answer is the first option? Wait, no, wait the question is which rule is violated. Wait, the three conditions for continuity: the first condition is that \( f(a) \) is defined. If \( f(a) \) is not defined, then the first condition is violated. So at \( x = 2 \), since there's an open circle, \( f(2) \) is not defined, so the rule " \( f(a) \) is defined" is violated.

Answer:

\( f(a) \) is defined. (The rule that \( f(a) \) is defined is violated at \( x = 2 \) because there is an open circle at \( x = 2 \), meaning \( f(2) \) is not defined. The limit as \( x \to 2 \) exists (left - hand and right - hand limits are equal), so the other two rules are not violated (or the third rule can't be checked as the first rule is violated).)