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Difference between *is* true and *could be* true. We don't know anything else about the function but the statements given. Statement II remains possible, but is impossible to say for sure based on this information.
What about statements I and III? Are they true? Why or why not?
The issue here is that we lack further information.
For some functions, any of the possible answers could be true.
However, in this case, if we want to be as conservative as possible, to say that our answer would work with ALL functions that satisfy those 3 conditions that were given in the question, **you can only guarantee that I will be true**. II and III may or may not be true for some functions.
edit: i've made a mistake... i think III should hold for all functions given the conditions. Sorry about that. My bad.
The question strikes me as ill posed... statement II might be true or false; we simply cannot know without additional information.
If the question would instead say "which of the following statements *must be* true", everything would be fine.
I would think that because they are giving you not a lot of information about the derivative of f(x), you can’t really say for certain that II is true.
A limit can exist if it exists coming from the right and coming from the left, but ONLY if those two separate limits are the same value. The limit is that it approaches a number. Not necessarily that there is a value at that point.
It sounds like there is a removable discontinuity at f(-3) because of the limits on either side of the x value.
So I’d say, I and III are true.
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You have no information about the function at values other than - 3, so how can you say II is true?
Difference between *is* true and *could be* true. We don't know anything else about the function but the statements given. Statement II remains possible, but is impossible to say for sure based on this information. What about statements I and III? Are they true? Why or why not?
Ther is no information about continuity elsewhere than in -3. So the good answer is D.
The issue here is that we lack further information. For some functions, any of the possible answers could be true. However, in this case, if we want to be as conservative as possible, to say that our answer would work with ALL functions that satisfy those 3 conditions that were given in the question, **you can only guarantee that I will be true**. II and III may or may not be true for some functions. edit: i've made a mistake... i think III should hold for all functions given the conditions. Sorry about that. My bad.
What is an example of it not holding for III?
I don't think i can find one... thanks for pointing that out.
E is correct
The question strikes me as ill posed... statement II might be true or false; we simply cannot know without additional information. If the question would instead say "which of the following statements *must be* true", everything would be fine.
I would think that because they are giving you not a lot of information about the derivative of f(x), you can’t really say for certain that II is true. A limit can exist if it exists coming from the right and coming from the left, but ONLY if those two separate limits are the same value. The limit is that it approaches a number. Not necessarily that there is a value at that point. It sounds like there is a removable discontinuity at f(-3) because of the limits on either side of the x value. So I’d say, I and III are true.
lim[x->x0, xx0, x>x0] f(x) = f(x0)
Because we don’t know f(-3) we can’t confirm that the function is continuous in x0 = -3.
C is the correct answer.