I By Continuity definition 1/0 is infinity

1. Mar 16, 2017

NotASmurf

lim 1/x as x->0 is infinity, but the function taking it to infinity is continuous, but for continous functions f(a)= lim f(x) as x->a, so by defininition 1/0 is infinity, what is wrong with this logic?

2. Mar 16, 2017

pwsnafu

1/x is not defined at x=0. This means that $f(a)$ does not exist, hence you can't appeal to continuity (continuity requires a function to be defined there).

3. Mar 16, 2017

NotASmurf

a is 0, f=1/x

4. Mar 16, 2017

pwsnafu

Yeah, sorry. I realised and edited my post. You replied just before I finished.

5. Mar 16, 2017

NotASmurf

oh, 1/x, x element of R, and inf not element of R but cardinality, thanks.

6. Mar 16, 2017

pwsnafu

Also, I want to point out
is not true. The left and right limits are not the same.

7. Mar 16, 2017

NotASmurf

Why not? 1/x coming from the negative side to 0 should yield same result as from the positive?

8. Mar 16, 2017

PeroK

No.The limit from the left is $-\infty$. For example $1/(-.01) = -100$

9. Mar 16, 2017

Staff: Mentor

There is no way to extend the definition of f to x=0 in a continuous way.

10. Mar 16, 2017

Staff: Mentor

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 $\lim_{x \to 0^+} \frac 1 x = +\infty$.