1.6.365 AP Calculus Exam Limits

[karush]

ok I chose c being false, since a limit does not exist if f(x) is different coming from $\pm$

 

[Greg]

You need to look at left-hand and right-hand limits. I don't see where "$\pm$" comes into it.

 

[HallsofIvy]

I believe that "from the left and from the right" was what Karush meant by "$$\pm$$!

Yes, Karush, "c" is the only one that is false.

a) $$\lim_{x\to 2} f(x)$$ exists.
True. The limit is 2. (f(2)= 1 so f is NOT continuous there.)

b) $$\lim_{x\to 3} f(x)$$ exists.
True. The limit is 5. (Further f(3)= 5 so f is continuous there.)

c) $$\lim_{x\to 4} f(x)$$ exists.
False. The "limit from the left", $$\lim_{x\to 4^-} f(x)$$, is 2 while the "limit from the right", $$\lim_{x\to 4^+} f(x)$$, is 4. Since the two one-sided limits are not the same the limit itself does not exist.

d) $$\lim_{x\to 5} f(x)$$ exists.
True. The limit is 6. (Further f(5)= 6 so f is continuous there.)

e) f is continuous at x= 3.
True. As I said in (b), $$\lim_{x\to 3} f(x)$$ and f(3) both exist and are equal.

 

[karush]

ok I think the big visual hint on this one was the obvious disconnected gap.

mahalo everyone the comments really increase the insight on these.