Does [1 - x] equal to -1 and why?

Steven G

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Can you show your work so that a helper on this forum knows what type of help you need.
 

Dr.Peterson

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As pka said, you'll need to define what you mean by that. It seems likely, from the nature of the problem, that you mean what is shown here, which is the greatest integer, or floor, function.

If so, then can you rewrite the entire function for x slightly greater than 1, and for x slightly less than 1?
I should add that the limit is not 1 by this interpretation; is there a typo? It wouldn't take a big change.
 

38175425

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Sorry for not being clear enough.

My work:
mywork.jpg

Actual complete question:
translator.png
translation:
右極限 = right-hand limit
左極限 = left-hand limit
想像 = imagine
故 = therefore
 

Steven G

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Let me be sure that I am understanding you.
1) The left hand limit = 2
2) The right hand limit = 2
3) Conclude that the limit is 1

Is that what you are saying??
 

Dr.Peterson

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Sorry for not being clear enough.

My work:
View attachment 34553

Actual complete question:
View attachment 34554
translation:
右極限 = right-hand limit
左極限 = left-hand limit
想像 = imagine
故 = therefore
If the left-hand and right-hand limits are both 2, why would the limit be 1??

And what parts of this do you not understand? Your question implied you don't understand any of it. Do you see how my hint explains what they did?
 

Steven G

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Let me be sure that I am understanding you.
1) The left hand limit = 2
2) The right hand limit = 2
3) Conclude that the limit is 1

Is that what you are saying??
You seriously need to learn how to tell limits from a graph. If you knew this, then you would never say
1) The left hand limit = 2
2) The right hand limit = 2
3) Conclude that the limit is 1
 

Dr.Peterson

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Problem solved thank you guys!
We would be more sure that you understand if you told us in what sense the problem is solved. Do you understand that the final answer given there is incorrect, though the work is correct? Do you understand how each part of the limit works -- namely, that for x between 0 and 1, the expression simplifies to 2+0+0=2, and for x between 1 and 2 it simplifies to 2+1-1=2, so that the limit on each side is 2?

The work you showed doesn't take this into account, and seems to be working backward from the solution you were given.
 

38175425

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We would be more sure that you understand if you told us in what sense the problem is solved. Do you understand that the final answer given there is incorrect, though the work is correct? Do you understand how each part of the limit works -- namely, that for x between 0 and 1, the expression simplifies to 2+0+0=2, and for x between 1 and 2 it simplifies to 2+1-1=2, so that the limit on each side is 2?

The work you showed doesn't take this into account, and seems to be working backward from the solution you were given.
The question and the final answer I posted are from a textbook. Maybe I'll have to ask my math teacher if the textbook did it wrong. Thanks for your explanation.
 

Steven G

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\(\displaystyle \lim_{x->a}f(x) = L\) if and only if the following two conditions are met.
1) \(\displaystyle \lim_{x->a^-}f(x) = L\)
2) \(\displaystyle \lim_{x->a^+}f(x) = L\)
 

Dr.Peterson

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I think the answer might be a typo, it should be 2.
Yes, that's what I'm suggesting.

Incidentally, here is a graph of the expression [imath]y=2+\operatorname{floor}\left(x\right)+\operatorname{floor}\left(1-x\right)[/imath] on Desmos:

1669569587039.png

I think that makes the limit rather clear.
 

jonah2.0

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