Which of the following functions is continuous at x = 3? (5 points)

Select one:
a. f of x equals the quotient of the quantity x squared minus 9 and the quantity x plus 3
b. f of x equals the quotient of the quantity x squared minus 9 and the quantity x minus 3 for x not equal to 3 and equals 3 for x equals 3
c. f of x equals the quotient of the quantity x squared minus 9 and the quantity x minus 3 for x not equal to 3 and equals 0 for x equals 3 Incorrect
d. All are continuous at x = 3

(a) f is continuous at all points except x = -3, so it is continuous at x=3

a) f(x) = (x^2 - 9)/(x+3)

= (x^2 - 3^2)/(x+3)
= (x+3)(x-3)/(x+3)
= (x-3)

Since this function is a linear one, it'll be continuous at all points.

b) Using the similar identity as above,
f(x) = (x+3), for x =/= 3
Now, at all points except 3, the function is given by (x+3). So at points getting closer to 3, the function tends towards 6. But as per the question, f(3) = 3.
Hence, it is discontinous at 3.

c) This is the same as part (b), except f(3) equals 0 instead of 3. However, the same argument can be used to show its discontinuity.

d. All are continuous at x = 3

Well, isn't it nice when all the options get along and agree to be continuous at x=3? It's like they joined hands and formed a never-ending line of smooth, seamless functions. Let's give a round of applause to these well-behaved functions!

To determine which function is continuous at x = 3, we need to check if all three conditions for continuity are met:

1. The function must be defined at x = 3.
2. The limit of the function as x approaches 3 must exist.
3. The value of the function at x = 3 must be equal to the limit.

Let's evaluate each function based on these conditions:

a. f(x) = (x^2 - 9)/(x + 3)
We can observe that this function is defined for all values of x, including x = 3. Now we need to check the limit:

lim(x->3) [(x^2 - 9)/(x + 3)]
Using direct substitution, we get:
lim(x->3) [(3^2 - 9)/(3 + 3)] = 0/6 = 0

Since the limit exists and is equal to 0, we move to the next condition. Let's evaluate the function at x = 3:

f(3) = (3^2 - 9)/(3 + 3) = 0

Since the value of the function at x = 3 is equal to the limit, function a is continuous at x = 3.

b. f(x) = (x^2 - 9)/(x - 3) for x ≠ 3, and f(x) = 3 for x = 3
In this case, the function is not defined at x = 3, so it fails the first condition and is not continuous at x = 3.

c. f(x) = (x^2 - 9)/(x - 3) for x ≠ 3, and f(x) = 0 for x = 3
Similar to function b, this function is not defined at x = 3, so it also fails the first condition and is not continuous at x = 3.

d. All three functions mentioned above are continuous at x = 3.
This statement is incorrect since functions b and c are not continuous at x = 3.

Therefore, the correct answer is a. f(x) = (x^2 - 9)/(x + 3) is continuous at x = 3.

To determine which of the given functions is continuous at x = 3, we need to check if the function is defined at x = 3 and if the limit of the function as x approaches 3 exists and is equal to the value of the function at x = 3.

Let's examine each option to determine if it meets these criteria:

a. f(x) = (x^2 - 9) / (x + 3)
To check continuity at x = 3, we need to ensure that the function is defined at x = 3 and that the limit of the function as x approaches 3 exists and is equal to the value of the function at x = 3.
Evaluating the function at x = 3, we get f(3) = (3^2 -9) / (3 + 3) = 0/6 = 0.
To check the limit as x approaches 3, we need to calculate the limit as x approaches 3 of f(x). Taking the limit:
lim (x -> 3) [(x^2 - 9) / (x + 3)] = (3^2 - 9) / (3 + 3) = 0/6 = 0.
Since the function is defined at x = 3, and the limit as x approaches 3 exists and is equal to the value of the function at x = 3, function a is continuous at x = 3.

b. f(x) = (x^2 - 9) / (x - 3) for x ≠ 3, and f(3) = 3
In this function, at x = 3, f(3) = 3, which is defined. However, we still need to check the limit as x approaches 3 for continuity.
lim (x -> 3) [(x^2 - 9) / (x - 3)] can be evaluated using direct substitution: lim (x -> 3) [3(x + 3)] / (x - 3) = 18 / 0, which is undefined.
Since the limit does not exist, function b is not continuous at x = 3.

c. f(x) = (x^2 - 9) / (x - 3) for x ≠ 3, and f(3) = 0
Based on the same reasoning, this function will also not be continuous at x = 3 because the limit as x approaches 3 will be undefined.

From the options provided, only function a is continuous at x = 3. Therefore, the correct answer is a.