What about
Hi,
First post, was not sure where to start with this one. I'm looking for an example of a function that is defined for all real numbers and for which the limit as x approaches 0 exists, but the limit as x approaches 0 is not the same as when the function f(0).
IIRC this would mean it is discontinuous (removable I think?) so would any function that when direct substitution is used returns 0/0 (and requires factoring, common denominator, conjugate, etc to find its limit) work for this? or would that still not be defined for all real numbers I guess, I'm kind of lost on where to start.
Thanks for the response, this is where I get confused though - the first one would be 2/0 and dividing by 0 makes it undefined, so would that disqualify the "function that is defined for all numbers" part? Also x would be approaching 0, not -1, unless i'm looking at something differently here?
It's a hybrid function, we have realised that the first part of the function is undefined at the point x = -1 so we have defined the function to have a value at that point.
And of course, you need to evaluate the limit as x goes to -1, because that's the point of discontinuity.
Is -1 coming from anything specific though or is it just a random number? I know it makes the first 0, but using 0 itself does too, so is there any other reason why -1 is used?
Sorry, just having trouble wrapping my head around these concepts in general.