Is it possible to have $$\lim_{x\to 1} f(x)=-23$$ and $f(1)=107$? I just don't know how to explain this and I don't even know if the above situation is possible. Please help me.
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Yes, if $f$ is not continuous you can construct those situations. – Jakob Elias Feb 03 '17 at 09:07
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Can u please be a bit more clear? – Yami Kanashi Feb 03 '17 at 09:08
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2Define a function $f$ that is constant $f(x)=-23$ for $x < 1$ and $f(1)=107$. Then the left-sided limit of $f$ when $x$ approaches $1$ is $-23$ but its value is $107$ at $x=1$. – Jakob Elias Feb 03 '17 at 09:11
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Ok I got it. Thank you. – Yami Kanashi Feb 03 '17 at 09:12
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You're welcome. – Jakob Elias Feb 03 '17 at 09:13
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The answer is yes.
The most simple function like that is:
$$f(x)=-23\quad \text{for $x\ne 1$ and } f(1)=107$$
Arnaldo
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Nice! But I have another doubt. If lim x->-17 f(x)=20, is it possible to determine f(-17)? – Yami Kanashi Feb 03 '17 at 09:23
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If you have just that condition then NO. You must have some other condition about $f$. The above answer cans help. I can define $f(1)$ as I want, for example it could be $f(1)=0$ instead of $107$. – Arnaldo Feb 03 '17 at 09:27
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It's only the condition that I have specified in the above comment – Yami Kanashi Feb 03 '17 at 09:28
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Oh good! So the reason is that sufficient properties of f are not mentioned, right? – Yami Kanashi Feb 03 '17 at 09:30
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