A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

Babynini

  • one year ago

More limits :)

  • This Question is Closed
  1. Babynini
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    #7

    1 Attachment
  2. jim_thompson5910
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    \[ |x| = \begin{cases} x \ \text{ if } \ x \ge 0\\ -x \ \text{ if } \ x < 0\\ \end{cases} \] \[ |\color{red}{x}| = \begin{cases} (\color{red}{x}) \ \text{ if } \ (\color{red}{x}) \ge 0\\ -(\color{red}{x}) \ \text{ if } \ (\color{red}{x}) < 0\\ \end{cases} \] \[ |\color{red}{2x+1}| = \begin{cases} (\color{red}{2x+1}) \ \text{ if } \ (\color{red}{2x+1}) \ge 0\\ -(\color{red}{2x+1}) \ \text{ if } \ (\color{red}{2x+1}) < 0\\ \end{cases} \] \[ |2x+1| = \begin{cases} 2x+1 \ \text{ if } \ x \ge -\frac{1}{2}\\ -2x-1 \ \text{ if } \ x < -\frac{1}{2}\\ \end{cases} \] based on the last definition above, we can replace the `|2x+1|` with just `2x+1` because x is getting closer and closer to 0 (0 makes x >= -1/2 true)

  3. jim_thompson5910
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Similarly, \[ |2x-1| = \begin{cases} 2x-1 \ \text{ if } \ x \ge \frac{1}{2}\\ -2x+1 \ \text{ if } \ x < \frac{1}{2}\\ \end{cases} \] So `|2x-1|` can be replaced with `-2x+1` (x=0 makes x < 1/2 true)

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

    How did you come up with the 1/2 's ?

  5. jim_thompson5910
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    solving 2x+1 >= 0, 2x+1 < 0, etc etc. The restrictions placed on each piece of the piecewise function

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

    ah k gotcha.

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

    So now we replace the values in the original with the ones you've just come up with.

  8. jim_thompson5910
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    correct

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

    Just a sec.

  10. jim_thompson5910
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    ok

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

    So it's [(2x+1)-(-2+1)]/x

  12. jim_thompson5910
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    close, there's an x missing

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

    Right on the second 2, sorry xD

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

    But that al simplifies to 4x/x?

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

    which of course = 4

  16. jim_thompson5910
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Sorry you should have this \[ \lim_{x\to 0} \frac{|2x-1|-|2x+1|}{x} = \lim_{x\to 0} \frac{(-2x+1)-(2x+1)}{x} \]

  17. jim_thompson5910
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    you somehow mixed up |2x-1| and |2x+1|

  18. jim_thompson5910
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    It should be -4. The graph confirms it https://www.desmos.com/calculator/4uqoaaryhi Definitely an odd and interesting graph

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

    haha looks...cool?

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

    okay, I got it right now :) so the entire limit = - 4

  21. jim_thompson5910
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    correct, as x gets closer and closer to 0 (from either side) the value of y gets closer and closer to -4

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

    Fabulous. Thank you so much. So at the end I would just rewrite the original limit and then put it all equals -4? correct?

  23. jim_thompson5910
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    yeah after you simplify you'll get -4x/x = -4 then you just apply the limit \[\Large \lim_{x\to 0}(-4)\] to get -4 itself

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

    oh! lovely!

  25. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

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.