Trying to understand a concept where the limits are as x approaches a constant, lets go with 2, is it possible to have a function where f(2) = 0 but have the limit not be 0. I've tried to create a variety of functions where in which case f(2) will = 0 but in turn this results in the limit also being 0.
Limits approaching a constant: can there exist f(x) so f(2)=0, but limit isn't 0?
- Thread starter bobman25
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HallsofIvy
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Consider f(x)= 3 for any x except 2 and f(2)= 0.
If \(\displaystyle \lim_{x\to 2} f(x)= f(2)\) then f is continuous at x= 2. While continuous functions are very important, almost all functions (in a very precise sense) are not continuous.
If \(\displaystyle \lim_{x\to 2} f(x)= f(2)\) then f is continuous at x= 2. While continuous functions are very important, almost all functions (in a very precise sense) are not continuous.
you can do this, but you need a removable discontinuity. Which is basically like a piece wise function that would look something like
f(x) = 1
f(2) = 0
here is an example:
I'm sorry for the alignment. I can't figure out how to flip the image. You can see, both of these functions have limits different than their values at that point.
f(x) = 1
f(2) = 0
here is an example:
I'm sorry for the alignment. I can't figure out how to flip the image. You can see, both of these functions have limits different than their values at that point.