Question

determine whether the following function is continuous at a. use the continuity checklist to justify your answer. f(x) = \frac{6x - 5}{x^2 - 11x + 30}, a = 5. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. a. the function is not continuous at a = 5 because although f(5) = \boxed{} is defined and \lim_{x\to 5} f(x) = \boxed{} exists, f(5) \
eq \lim_{x\to 5} f(x). b. the function is continuous at a = 5 because \lim_{x\to 5} f(x) = \boxed{} exists. c. the function is continuous at a = 5 because f(5) = \boxed{} is defined. d. the function is continuous at a = 5 because f(5) = \boxed{} is defined and \lim_{x\to 5} f(x) exists and is equal to f(5). e. the function is not continuous at a = 5 because although f(5) = \boxed{} is defined, \lim_{x\to 5} f(x) does not exist. f. the function is not continuous at a = 5 because f(5) is not defined.

Explanation:

Step1: Factor the denominator

First, factor the quadratic in the denominator: \(x^2 - 11x + 30\). We need two numbers that multiply to 30 and add to -11. Those numbers are -5 and -6. So, \(x^2 - 11x + 30=(x - 5)(x - 6)\). So the function becomes \(f(x)=\frac{6x - 5}{(x - 5)(x - 6)}\).

Step2: Check if \(f(5)\) is defined

To find \(f(5)\), substitute \(x = 5\) into the function. The denominator becomes \((5 - 5)(5 - 6)=0\times(-1)=0\). Division by zero is undefined, so \(f(5)\) is not defined.

Step3: Determine continuity

For a function to be continuous at a point \(a\), three conditions must be met:

1. \(f(a)\) is defined.
2. \(\lim_{x

ightarrow a}f(x)\) exists.

1. \(\lim_{x

ightarrow a}f(x)=f(a)\).

Since \(f(5)\) is not defined (fails the first condition), the function is not continuous at \(a = 5\) because \(f(5)\) is not defined.

Answer:

F. The function is not continuous at \(a = 5\) because \(f(5)\) is not defined.