## ParthKohli Group Title Is there a removable discontinuity when a limit exists? one year ago one year ago

1. abb0t Group Title

a removable discontinuity is one at which the limit of the function exists but does not equal the value of the function at that point; this may be because the function does not exist at that point. |dw:1356113778802:dw|

2. ParthKohli Group Title

What if there's a hole at a point and the limit *is* the hole? What discontinuity is that called?

3. TuringTest Group Title

...but whether or not the unction value exists, for a removeable discontinuity we have that$\lim_{x\to a^1}f(x)=\lim_{x\to a^+}f(x)\neq f(a),\pm\infty$i.e. the limit must exist for a removeable discontinuity.

4. abb0t Group Title

Step Continuity?

5. ParthKohli Group Title

Oh, I get it.

6. TuringTest Group Title

the thing you describe parth is a removable discontinuity

7. ParthKohli Group Title

Ah, okay :)

8. TuringTest Group Title

the limit exists (is finite) but is not the same as the value of the function at that point. the actual value of the function at that point can be anything... even f(a)=infinity, it would still be removable

9. ParthKohli Group Title

Got it, so the discontinuity in a function$f(x) = {{\rm foo} \over x + 2}$is a removable discontinuity right?

10. TuringTest Group Title

no, because $\lim_{x\to0^-}f(x)\neq\lim_{x\to0^+}f(x)$

11. TuringTest Group Title

I mean $$x\to-2$$

12. TuringTest Group Title

also, both limits do not exist; for a removable discontinuity the limit at the discontinuity must exist ( be finite and equal on both sides)

13. TuringTest Group Title

here $$x\to2^-\implies f(x)\to\infty,~~~x\to2^+\implies f(x)\to-\infty$$

14. ParthKohli Group Title

Ah! Okay, so the limit doesn't exist.

15. TuringTest Group Title

so the limits are neither equal on both sides, or even the same sign!

16. TuringTest Group Title

correct, for many reasons it doesn't exist

17. TuringTest Group Title

it is an essential discontinuity

18. ParthKohli Group Title

hmm

19. TuringTest Group Title

|dw:1356114605529:dw|

20. ParthKohli Group Title

I see :)

21. TuringTest Group Title

as $$x\to-2^+$$ we are going this way off to infinity|dw:1356114734440:dw|

22. ParthKohli Group Title

Yup, I see it now :) thanks

23. TuringTest Group Title

that alone is enough to say that the discontinuity is essential, and that the limit does not exist at x=-2 but still we have the other argument as well. from the left we have the opposite case|dw:1356114780902:dw|

24. ParthKohli Group Title

Aww, no one cares about the vertical asymptote $$x = -2$$... give it some love :)

25. TuringTest Group Title

forever alone I guess :P

26. ParthKohli Group Title

lol

27. ParthKohli Group Title

Thanks @TuringTest :)

28. TuringTest Group Title

welcome :D