Question

determine whether the following function is continuous at ( a ). use the continuity checklist to justify your answer.
f(x) = \begin{cases} dfrac{x^2 - 81}{x - 9} & \text{if } x
eq 9 \\ 8 & \text{if } x = 9 end{cases}; , a = 9
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 = 9 ) because ( f(9) ) is not defined.

b. the function is not continuous at ( a = 9 ) because although ( f(9) = \boxed{} ) is defined and ( limlimits_{x \to 9} f(x) = \boxed{} ) exists, ( f(9)
eq limlimits_{x \to 9} f(x) )

c. the function is continuous at ( a = 9 ) because ( limlimits_{x \to 9} f(x) = \boxed{} ) exists.

d. the function is continuous at ( a = 9 ) because ( f(9) = \boxed{} ) is defined.

e. the function is not continuous at ( a = 9 ) because although ( f(9) = \boxed{} ) is defined, ( limlimits_{x \to 9} f(x) ) does not exist.

f. the function is continuous at ( a = 9 ) because ( f(9) = \boxed{} ) is defined and ( limlimits_{x \to 9} f(x) ) exists and is equal to ( f(9) ).

Explanation:

Step1: Simplify the function for \(x

eq9\)
We have the function \(f(x)=\frac{x^{2}-81}{x - 9}\) for \(x
eq9\). Notice that \(x^{2}-81\) is a difference of squares, so we can factor it as \((x - 9)(x + 9)\). Then \(\frac{x^{2}-81}{x - 9}=\frac{(x - 9)(x + 9)}{x - 9}\). Since \(x
eq9\), we can cancel out the \((x - 9)\) terms, and we get \(f(x)=x + 9\) for \(x
eq9\).

Step2: Find the limit as \(x\to9\)

To find \(\lim_{x\to9}f(x)\), we use the simplified function \(f(x)=x + 9\) (for \(x
eq9\)). So \(\lim_{x\to9}(x + 9)=9+9 = 18\).

Step3: Find \(f(9)\)

From the definition of the function, when \(x = 9\), \(f(9)=8\).

Step4: Check the continuity conditions

For a function to be continuous at \(a = 9\), we need \(\lim_{x\to9}f(x)=f(9)\). We found that \(\lim_{x\to9}f(x)=18\) and \(f(9)=8\). Since \(18
eq8\), the function is not continuous at \(a = 9\) because although \(f(9) = 8\) is defined and \(\lim_{x\to9}f(x)=18\) exists, \(f(9)
eq\lim_{x\to9}f(x)\). So the correct option is B. And the value of \(\lim_{x\to9}f(x)\) is \(18\) and \(f(9)=8\).

Answer:

B. The function is not continuous at \(a = 9\) because although \(f(9)=8\) is defined and \(\lim_{x\to9}f(x)=18\) exists, \(f(9)
eq\lim_{x\to9}f(x)\)