determine whether the following function is continuous at a. use the continuity checklist to justify your…

determine whether the following function is continuous at a. use the continuity checklist to justify your answer. f(x) = { (x^2 - 36)/(x - 6) if x≠6; 3 if x = 6 ; a = 6 select all that apply. a. the function is continuous at a = 6. b. the function is not continuous at a = 6 because f(6) is undefined. c. the function is not continuous at a = 6 because lim(x→6) f(x) does not exist. d. the function is not continuous at a = 6 because lim(x→6) f(x)≠f(6).

determine whether the following function is continuous at a. use the continuity checklist to justify your answer. f(x) = { (x^2 - 36)/(x - 6) if x≠6; 3 if x = 6 ; a = 6 select all that apply. a. the function is continuous at a = 6. b. the function is not continuous at a = 6 because f(6) is undefined. c. the function is not continuous at a = 6 because lim(x→6) f(x) does not exist. d. the function is not continuous at a = 6 because lim(x→6) f(x)≠f(6).

Answer

Explanation:

Step1: Simplify the function for $x\neq6$

For $x\neq6$, $f(x)=\frac{x^{2}-36}{x - 6}=\frac{(x + 6)(x - 6)}{x - 6}=x + 6$.

Step2: Calculate the limit as $x\to6$

$\lim_{x\to6}f(x)=\lim_{x\to6}(x + 6)=6+6 = 12$.

Step3: Evaluate the function at $x = 6$

$f(6)=3$.

Step4: Check the continuity condition

Since $\lim_{x\to6}f(x)=12$ and $f(6)=3$, $\lim_{x\to6}f(x)\neq f(6)$.

Answer:

D. The function is not continuous at a = 6 because $\lim_{x\to6}f(x)\neq f(6)$.