Quantcast

Got Homework?

Connect with other students for help. It's a free community.

  • across
    MIT Grad Student
    Online now
  • laura*
    Helped 1,000 students
    Online now
  • Hero
    College Math Guru
    Online now

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

ParthKohli

Is there a removable discontinuity when a limit exists?

  • one year ago
  • one year ago

  • This Question is Closed
  1. abb0t
    Best Response
    You've already chosen the best response.
    Medals 0

    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|

    • one year ago
  2. ParthKohli
    Best Response
    You've already chosen the best response.
    Medals 2

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

    • one year ago
  3. TuringTest
    Best Response
    You've already chosen the best response.
    Medals 1

    ...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.

    • one year ago
  4. abb0t
    Best Response
    You've already chosen the best response.
    Medals 0

    Step Continuity?

    • one year ago
  5. ParthKohli
    Best Response
    You've already chosen the best response.
    Medals 2

    Oh, I get it.

    • one year ago
  6. TuringTest
    Best Response
    You've already chosen the best response.
    Medals 1

    the thing you describe parth is a removable discontinuity

    • one year ago
  7. ParthKohli
    Best Response
    You've already chosen the best response.
    Medals 2

    Ah, okay :)

    • one year ago
  8. TuringTest
    Best Response
    You've already chosen the best response.
    Medals 1

    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

    • one year ago
  9. ParthKohli
    Best Response
    You've already chosen the best response.
    Medals 2

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

    • one year ago
  10. TuringTest
    Best Response
    You've already chosen the best response.
    Medals 1

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

    • one year ago
  11. TuringTest
    Best Response
    You've already chosen the best response.
    Medals 1

    I mean \(x\to-2\)

    • one year ago
  12. TuringTest
    Best Response
    You've already chosen the best response.
    Medals 1

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

    • one year ago
  13. TuringTest
    Best Response
    You've already chosen the best response.
    Medals 1

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

    • one year ago
  14. ParthKohli
    Best Response
    You've already chosen the best response.
    Medals 2

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

    • one year ago
  15. TuringTest
    Best Response
    You've already chosen the best response.
    Medals 1

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

    • one year ago
  16. TuringTest
    Best Response
    You've already chosen the best response.
    Medals 1

    correct, for many reasons it doesn't exist

    • one year ago
  17. TuringTest
    Best Response
    You've already chosen the best response.
    Medals 1

    it is an essential discontinuity

    • one year ago
  18. ParthKohli
    Best Response
    You've already chosen the best response.
    Medals 2

    hmm

    • one year ago
  19. TuringTest
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1356114605529:dw|

    • one year ago
  20. ParthKohli
    Best Response
    You've already chosen the best response.
    Medals 2

    I see :)

    • one year ago
  21. TuringTest
    Best Response
    You've already chosen the best response.
    Medals 1

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

    • one year ago
  22. ParthKohli
    Best Response
    You've already chosen the best response.
    Medals 2

    Yup, I see it now :) thanks

    • one year ago
  23. TuringTest
    Best Response
    You've already chosen the best response.
    Medals 1

    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|

    • one year ago
  24. ParthKohli
    Best Response
    You've already chosen the best response.
    Medals 2

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

    • one year ago
  25. TuringTest
    Best Response
    You've already chosen the best response.
    Medals 1

    forever alone I guess :P

    • one year ago
  26. ParthKohli
    Best Response
    You've already chosen the best response.
    Medals 2

    lol

    • one year ago
  27. ParthKohli
    Best Response
    You've already chosen the best response.
    Medals 2

    Thanks @TuringTest :)

    • one year ago
  28. TuringTest
    Best Response
    You've already chosen the best response.
    Medals 1

    welcome :D

    • one year ago
    • Attachments:

See more questions >>>

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.