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mathcalculus

  • 2 years ago

@asnaseer finding the limit:

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  1. mathcalculus
    • 2 years ago
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  2. asnaseer
    • 2 years ago
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    have you attempted to sketch the curve first?

  3. FutureMathProfessor
    • 2 years ago
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    |dw:1375034315072:dw|

  4. asnaseer
    • 2 years ago
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    which part are you stuck on?

  5. mathcalculus
    • 2 years ago
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    how do i sketch if i dont know the points

  6. mathcalculus
    • 2 years ago
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    all i see are x+1, x <> 2 then im stuck

  7. mathcalculus
    • 2 years ago
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    i can't use calculator.

  8. asnaseer
    • 2 years ago
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    ok - lets break it down into smaller steps

  9. asnaseer
    • 2 years ago
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    first, you are told that:\[f(x)=x+1\]if \(x\le2\) agreed?

  10. mathcalculus
    • 2 years ago
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    yes

  11. asnaseer
    • 2 years ago
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    so lets sketch this...

  12. mathcalculus
    • 2 years ago
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    so for a) since it;s coming from the left side it must be negative... and we use the top function, correct?

  13. FutureMathProfessor
    • 2 years ago
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    Its already been sketched btw

  14. asnaseer
    • 2 years ago
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    |dw:1375034583963:dw|

  15. mathcalculus
    • 2 years ago
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    ok how did you sketch that ? :(

  16. asnaseer
    • 2 years ago
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    I just sketched y=x+1 got 2 points on it: when x=0, y=1 when x=2, y=3 and then just drew a line

  17. asnaseer
    • 2 years ago
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    we are told that this is only for x less than or equal to 2. so we need to CHOP the curve at x=2 and draw a filled-in dot at that point. it needs to be filled-in because it INCLUDES the value at x=2

  18. asnaseer
    • 2 years ago
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    so the actual line should look like this:

  19. asnaseer
    • 2 years ago
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    does that make sense?

  20. mathcalculus
    • 2 years ago
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    ohhhhh okay :)

  21. asnaseer
    • 2 years ago
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    |dw:1375034970995:dw|

  22. mathcalculus
    • 2 years ago
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    so when are the times that we fill in the dots?

  23. mathcalculus
    • 2 years ago
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    and don't fill in?

  24. asnaseer
    • 2 years ago
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    you always fill-in the dots if the point INCLUDES the value - so all these would result in filling in the dot:\[x\le2\]\[x=2\]\[x\ge2\]and all these would NOT be filled-in:\[x<2\]\[x>2\]

  25. asnaseer
    • 2 years ago
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    note that the first 3 INCLUDE the point x=2

  26. asnaseer
    • 2 years ago
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    if that makes sense, then we can move onto the next part...

  27. asnaseer
    • 2 years ago
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    ?

  28. mathcalculus
    • 2 years ago
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    yes :)

  29. mathcalculus
    • 2 years ago
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    so stoping at two we fill in

  30. asnaseer
    • 2 years ago
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    yes

  31. mathcalculus
    • 2 years ago
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    and whatever is greater than 2 we leave alone?

  32. mathcalculus
    • 2 years ago
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    i got it..

  33. asnaseer
    • 2 years ago
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    ok, so 2nd part of the curve is defined as:\[f(x)=1\]if \(x\gt2\) agreed?

  34. mathcalculus
    • 2 years ago
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    yes

  35. asnaseer
    • 2 years ago
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    so first lets sketch f(x) = 1

  36. asnaseer
    • 2 years ago
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    |dw:1375035727853:dw|

  37. mathcalculus
    • 2 years ago
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    wait, can't we sketch x=3 because it says x is GREATER than 2

  38. asnaseer
    • 2 years ago
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    now here we are told that this part is only valid for \(x\gt2\), so we need to draw a circle (not filled-in) at the point x=2 and throw away everything to the left of this point

  39. asnaseer
    • 2 years ago
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    |dw:1375035852984:dw|

  40. mathcalculus
    • 2 years ago
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    yeah but wait, im not sure why we are using 1 when we can use 3 since it says x>2

  41. asnaseer
    • 2 years ago
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    the value of f(x) = 1 for x greater than 2

  42. asnaseer
    • 2 years ago
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    that includes points such as x=2.000001, x=2.5, x=3, x=99, etc

  43. asnaseer
    • 2 years ago
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    i.e. ALL points to the right of (and NOT including) x=2

  44. asnaseer
    • 2 years ago
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    |dw:1375035980350:dw|

  45. mathcalculus
    • 2 years ago
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    oh okay so i can use 3.. great

  46. asnaseer
    • 2 years ago
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    so now we just combine both parts of the curve to get...

  47. asnaseer
    • 2 years ago
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    |dw:1375036030210:dw|

  48. asnaseer
    • 2 years ago
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    make sense?

  49. asnaseer
    • 2 years ago
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    |dw:1375036145535:dw|

  50. mathcalculus
    • 2 years ago
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    got it

  51. asnaseer
    • 2 years ago
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    good. now we can examine the lmits

  52. mathcalculus
    • 2 years ago
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    ok :)

  53. asnaseer
    • 2 years ago
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    the 1st one is limit as x tends to 2 from the negative side

  54. asnaseer
    • 2 years ago
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    so - look at the curves and see what you think the answer should be for this case

  55. asnaseer
    • 2 years ago
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    |dw:1375036264164:dw|

  56. mathcalculus
    • 2 years ago
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    1 zt 2^-

  57. mathcalculus
    • 2 years ago
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    at*

  58. mathcalculus
    • 2 years ago
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    why is it -1 and not 1?

  59. asnaseer
    • 2 years ago
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    ?

  60. mathcalculus
    • 2 years ago
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    i see it's coming from the left side and the answer is going to be negative.

  61. asnaseer
    • 2 years ago
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    why is it going to be negative?

  62. mathcalculus
    • 2 years ago
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    but where do i look, in the y?

  63. mathcalculus
    • 2 years ago
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    because it's coming from the left?

  64. asnaseer
    • 2 years ago
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    lets approach the limit in steps

  65. mathcalculus
    • 2 years ago
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    how do i look for the limit when it approaches to 2?

  66. asnaseer
    • 2 years ago
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    |dw:1375036393036:dw|

  67. asnaseer
    • 2 years ago
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    |dw:1375036417600:dw|

  68. asnaseer
    • 2 years ago
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    can you see now what the limit should be as x approaches 2 from the left-hand-side?

  69. mathcalculus
    • 2 years ago
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    so its 3?

  70. asnaseer
    • 2 years ago
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    perfect! well done! :)

  71. asnaseer
    • 2 years ago
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    now try to do the same for x approaching 2 from the right-hand-side

  72. mathcalculus
    • 2 years ago
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    right side of 2 it approaches at 1

  73. asnaseer
    • 2 years ago
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    bingo! you're a pro now! :)

  74. mathcalculus
    • 2 years ago
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    2 is undefined?

  75. asnaseer
    • 2 years ago
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    the /tricky/ part now is what is the limit as x approaches 2 (without being told whether its from the left or right side)?

  76. asnaseer
    • 2 years ago
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    yes!

  77. asnaseer
    • 2 years ago
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    I believe you have grokked this now! :)

  78. mathcalculus
    • 2 years ago
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    because, it's not coming from either side.

  79. mathcalculus
    • 2 years ago
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    omg thank you!!!! :))

  80. mathcalculus
    • 2 years ago
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    thank you so much!!

  81. asnaseer
    • 2 years ago
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    yw - I'm glad I was to explain it well enough for you to understand. :)

  82. mathcalculus
    • 2 years ago
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    i'll apply this to the rest when im doing a problem

  83. asnaseer
    • 2 years ago
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    perfect! - keep at it - you seem to have a natural ability at this. :)

  84. mathcalculus
    • 2 years ago
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    What if they tell us to calculate instead of sketching?

  85. mathcalculus
    • 2 years ago
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    do we simply substitute?

  86. mathcalculus
    • 2 years ago
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    @asnaseer

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

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