Question

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

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

\quad a = 3$
\

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

Explanation:

Step1: Simplify the function for \( x

eq 3 \)
Factor the numerator and denominator: \( x^2 - 3x = x(x - 3) \), \( x^2 - 9 = (x - 3)(x + 3) \). So for \( x
eq 3 \), \( f(x)=\frac{x(x - 3)}{(x - 3)(x + 3)}=\frac{x}{x + 3} \) (cancel \( x - 3 \) as \( x
eq3 \)).

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

Calculate \( \lim_{x\to3}f(x) \) using the simplified function: \( \lim_{x\to3}\frac{x}{x + 3}=\frac{3}{3 + 3}=\frac{1}{2} \).

Step3: Check \( f(3) \) and compare with the limit

\( f(3) = 1 \) (given). Now, \( \lim_{x\to3}f(x)=\frac{1}{2} \) and \( f(3)=1 \), so \( \lim_{x\to3}f(x)
eq f(3) \). Also, check left and right limits: since the function is \( \frac{x}{x + 3} \) near \( x = 3 \) (for \( x
eq3 \)), left limit \( \lim_{x\to3^-}\frac{x}{x + 3}=\frac{3}{6}=\frac{1}{2} \), right limit \( \lim_{x\to3^+}\frac{x}{x + 3}=\frac{3}{6}=\frac{1}{2} \), so \( \lim_{x\to3}f(x) \) exists (is \( \frac{1}{2} \)).

So the correct options:

• The first option: Left and right limits are equal (both \( \frac{1}{2} \)), so this is wrong.
• Second option: \( f(3) = 1 \) is defined, so this is wrong.
• Third option: \( f(3)=1 \) is defined, \( \lim_{x\to3}f(x)=\frac{1}{2} \) is finite, and they are not equal. This is correct.
• Fourth option: \( \lim_{x\to3}f(x) \) exists (as left and right limits are equal), so this is wrong.
• Fifth option: Since third is correct, this is wrong.

Answer:

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