Question

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a) is the function $f(x)=(x-10)(x-5)$ continuous at $x=5$
○ yes it is continuous
○ no it is not continuous
b) select all of the options below that are true.
□ no rules are violated, the function is continuous.
☑ the function is not defined at $x=5$.
□ the limit does not exist.
☑ the value of the limit does not equal the value of the function at 5

Explanation:

Step1: Check function definition at $x=5$

Substitute $x=5$ into $f(x)$:
$f(5)=(5-10)(5-5)=(-5)(0)=0$
The function is defined at $x=5$.

Step2: Evaluate limit at $x=5$

Compute $\lim_{x \to 5} (x-10)(x-5)$:
$\lim_{x \to 5} (x-10)(x-5)=(5-10)(5-5)=0$
The limit exists and equals 0.

Step3: Verify continuity condition

A function is continuous at $x=a$ if:

1. $f(a)$ is defined,
2. $\lim_{x \to a} f(x)$ exists,
3. $\lim_{x \to a} f(x)=f(a)$

Here, $f(5)=0$ and $\lim_{x \to 5} f(x)=0$, so all conditions are satisfied.

Answer:

a) Yes it is continuous
b) No rules are violated, the function is continuous.