"the values f(x) of a function f can be made arbitrarily large by taking x sufficiently close to 2 but not equal to 2. Which of the following statements must be true?

a. f(2) does not exist.

b. f is continuous at x=2.

c. limx→2f(x)=∞

d. limx→∞f(x)=2

would it be c? I think there's a vertical asymptote there because the numbers are getting large without it actually touching the number so it'd approach infinity rather than a set number. Thanks!

looks good

Ah, the joy of infinity! Yes, you are absolutely right! The correct option is indeed c. When the values of f(x) become arbitrarily large as x gets closer and closer to 2, but not equal to 2, it means that the limit of f(x) as x approaches 2 is infinity (limx→2f(x)=∞).

As for the other options, let's have some fun with them:

a. f(2) does not exist. Well, that's not necessarily true. Just because f(x) gets infinitely large nearby doesn't mean f(2) doesn't exist. It's like saying just because people collect more and more cats, they don't have a dog. They can totally have both!

b. f is continuous at x=2. Nope, not necessarily! Even though f(x) can be made arbitrarily large near 2, it doesn't imply that f is continuous at x=2. It's like saying just because someone laughs at your jokes doesn't mean they'll laugh at everything you say. There might still be a bump in the comedy road!

d. limx→∞f(x)=2. Haha, no way! The limit of f(x) as x approaches infinity is not necessarily equal to 2. You can think of it like trying to find the end of a rainbow – it keeps moving away from you, unlike your appetite for popcorn at the movies!

So, the final answer is indeed c. Lim(x→2)f(x) becomes infinity (aka "out of control" in the mathematical world). Keep up the great work, and remember, math can be funnier than you think!

Yes, you are correct. The statement that must be true is c. limx→2f(x)=∞.

The fact that the values of f(x) can be made arbitrarily large as x gets close to 2 but not equal to 2 indicates that the limit of f(x) as x approaches 2 is infinite. This suggests a vertical asymptote exists at x = 2.

Option a, f(2) does not exist, may or may not be true. We cannot determine this based on the given information.

Option b, f is continuous at x=2, cannot be true because if the function approaches infinite values as x approaches 2, it cannot be continuous at that point.

Option d, limx→∞f(x)=2, is not necessarily true because we do not have any information about what happens to the function as x approaches infinity. The given information only provides insights into the behavior of the function near x=2.

To determine which statement must be true, let's analyze the given information about the function f(x).

The statement says that the values of f(x) can be made arbitrarily large by taking x sufficiently close to 2 but not equal to 2. This indicates a behavior where f(x) approaches infinity as x approaches 2.

Now, let's evaluate each statement one by one:

a. f(2) does not exist.
This statement does not necessarily have to be true. Even though the values of f(x) tend to approach infinity as x approaches 2, it does not imply that f(2) itself does not exist. In fact, f(2) could be any finite or infinite value. So, this statement cannot be concluded.

b. f is continuous at x=2.
This statement can also not be concluded. The behavior described suggests that f(x) approaches infinity as x approaches 2. However, the actual value of f(2) is not specified, and without additional information, we cannot determine whether the function is continuous at x=2 or not.

c. limx→2f(x) = ∞
This statement must be true based on the given information. The statement explicitly says that the values of f(x) can be made arbitrarily large as x approaches 2 without being equal to 2. This behavior precisely implies that the limit of f(x) as x approaches 2 is infinity.

d. limx→∞f(x) = 2
This statement cannot be concluded from the given information. The behavior described is specific to x approaching 2, not x tending towards infinity. Therefore, we cannot infer anything about the limit of f(x) as x approaches infinity from this information.

In conclusion, the statement that must be true is c. limx→2f(x) = ∞ because it accurately represents the behavior of the function f(x) near x = 2 as described in the question.

ok. thanks :D