The discussion revolves around the interpretation of the limit of the function 1/x as x approaches 0, specifically addressing the implications of continuity and the definition of limits in this context. Participants explore whether 1/0 can be considered infinity based on continuity definitions and the behavior of the function near the discontinuity at x=0.
Participants generally disagree on the interpretation of the limit of 1/x as x approaches 0 and the implications of continuity. There is no consensus on whether 1/0 can be defined as infinity.
Limitations include the undefined nature of the function at x=0, the differing left and right limits, and the implications of continuity that are not universally accepted in this context.
Yeah, sorry. I realized and edited my post. You replied just before I finished.NotASmurf said:a is 0, f=1/x
is not true. The left and right limits are not the same.lim 1/x as x->0 is infinity
pwsnafu said:Also, I want to point out
is not true. The left and right limits are not the same.
No.The limit from the left is ##-\infty##. For example ##1/(-.01) = -100##NotASmurf said:Why not? 1/x coming from the negative side to 0 should yield same result as from the positive?
There is no way to extend the definition of f to x=0 in a continuous way.NotASmurf said:but for continuous functions f(a)= lim f(x) as x->a
Take a look at the graph of y = 1/x. There is the worst possible kind of discontinuity at x = 0, with ##\lim_{x \to 0^-} \frac 1 x = -\infty## and ##\lim_{x \to 0^+} \frac 1 x = +\infty##.NotASmurf said:lim 1/x as x->0 is infinity, but the function taking it to infinity is continuous, but for continuous functions f(a)= lim f(x) as x->a, so by defininition 1/0 is infinity, what is wrong with this logic?