Question

explain why the function is discontinuous at the given number a. (select all that apply.)
$f(x) = \

$$\begin{cases}\\frac{x^2 - 3x}{x^2 - 9} & \\text{if } x \ eq 3 \\\\ 1 & \\text{if } x = 3 \\end{cases}$$

$ $a = 3$

• $\lim\limits_{x \to 3} f(x)$ does not exist.
• $\lim\limits_{x \to 3^+} f(x)$ and $\lim\limits_{x \to 3^-} f(x)$ are finite, but are not equal.
• $f(3)$ is defined and $\lim\limits_{x \to 3} f(x)$ is finite, but they are not equal.
• $f(3)$ is undefined.
• none of the above

sketch the graph of the function.

Explanation:

Step1: Simplify the function for \( x

eq 3 \)
Factor the numerator and denominator: \( \frac{x^2 - 3x}{x^2 - 9} = \frac{x(x - 3)}{(x - 3)(x + 3)} \). Cancel \( x - 3 \) (since \( x
eq 3 \), \( x - 3
eq 0 \)): \( \frac{x}{x + 3} \).

Step2: Find the limit as \( x \to 3 \)

Compute \( \lim_{x \to 3} \frac{x}{x + 3} \). Substitute \( x = 3 \): \( \frac{3}{3 + 3} = \frac{1}{2} \).

Step3: Analyze \( f(3) \)

Given \( f(3) = 1 \).

Step4: Check continuity conditions

A function is continuous at \( a \) if \( \lim_{x \to a} f(x) = f(a) \). Here, \( \lim_{x \to 3} f(x) = \frac{1}{2} \) and \( f(3) = 1 \). So \( \lim_{x \to 3} f(x) \) is finite, \( f(3) \) is defined, but they are not equal. Also, \( \lim_{x \to 3^+} f(x) = \lim_{x \to 3^-} f(x) = \frac{1}{2} \) (so the second option is wrong), and \( f(3) \) is defined (so the fourth option is wrong), and the limit exists (so the first option is wrong).

1. Simplify \( f(x) \) for \( x

eq 3 \): \( \frac{x^2 - 3x}{x^2 - 9} = \frac{x(x - 3)}{(x - 3)(x + 3)} = \frac{x}{x + 3} \) (for \( x
eq 3 \)).

1. Find \( \lim_{x \to 3} f(x) \): Substitute \( x = 3 \) into \( \frac{x}{x + 3} \), giving \( \frac{3}{6} = \frac{1}{2} \).
2. Analyze \( f(3) \): Given \( f(3) = 1 \).
3. Continuity condition: A function is continuous at \( a \) if \( \lim_{x \to a} f(x) = f(a) \). Here, \( \lim_{x \to 3} f(x) = \frac{1}{2} \) and \( f(3) = 1 \), so they are not equal.

• \( \lim_{x \to 3} f(x) \) exists (so "lim \( f(x) \) does not exist" is wrong).
• \( \lim_{x \to 3^+} f(x) = \lim_{x \to 3^-} f(x) = \frac{1}{2} \) (so "lim \( f(x) \) from right and left are finite but not equal" is wrong).
• \( f(3) = 1 \) is defined (so "f(3) is undefined" is wrong).
• The correct reason is: \( f(3) \) is defined, \( \lim_{x \to 3} f(x) \) is finite, but they are not equal.

Answer:

• The correct option is: \( f(3) \) is defined and \( \lim_{x \to 3} f(x) \) is finite, but they are not equal.