The discussion centers on the mathematical concept of limits, specifically the limit of the function f(x) = 1/x as x approaches 0. Participants clarify that while the limit approaches infinity, the function is not defined at x=0, resulting in a discontinuity. The left-hand limit approaches negative infinity, while the right-hand limit approaches positive infinity, highlighting the inconsistency in the logic that suggests 1/0 equals infinity. Thus, the conclusion is that continuity cannot be applied at x=0 for the function 1/x.
PREREQUISITESStudents of calculus, mathematics educators, and anyone interested in understanding the nuances of limits and continuity in mathematical functions.
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?