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mathcalculus

@asnaseer finding the limit:

  • 8 months ago
  • 8 months ago

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

    • 8 months ago
  3. FutureMathProfessor
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    |dw:1375034315072:dw|

    • 8 months ago
  4. asnaseer
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    which part are you stuck on?

    • 8 months ago
  5. mathcalculus
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    how do i sketch if i dont know the points

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

    • 8 months ago
  7. mathcalculus
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    i can't use calculator.

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

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

    • 8 months ago
  10. mathcalculus
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    yes

    • 8 months ago
  11. asnaseer
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    so lets sketch this...

    • 8 months ago
  12. mathcalculus
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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?

    • 8 months ago
  13. FutureMathProfessor
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    Its already been sketched btw

    • 8 months ago
  14. asnaseer
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    |dw:1375034583963:dw|

    • 8 months ago
  15. mathcalculus
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    ok how did you sketch that ? :(

    • 8 months ago
  16. asnaseer
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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

    • 8 months ago
  17. asnaseer
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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

    • 8 months ago
  18. asnaseer
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    so the actual line should look like this:

    • 8 months ago
  19. asnaseer
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    does that make sense?

    • 8 months ago
  20. mathcalculus
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    ohhhhh okay :)

    • 8 months ago
  21. asnaseer
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    |dw:1375034970995:dw|

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

    • 8 months ago
  23. mathcalculus
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    and don't fill in?

    • 8 months ago
  24. asnaseer
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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:\[x<2\]\[x>2\]

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

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

    • 8 months ago
  27. asnaseer
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    ?

    • 8 months ago
  28. mathcalculus
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    yes :)

    • 8 months ago
  29. mathcalculus
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    so stoping at two we fill in

    • 8 months ago
  30. asnaseer
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    yes

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

    • 8 months ago
  32. mathcalculus
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    i got it..

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

    • 8 months ago
  34. mathcalculus
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    yes

    • 8 months ago
  35. asnaseer
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    so first lets sketch f(x) = 1

    • 8 months ago
  36. asnaseer
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    |dw:1375035727853:dw|

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

    • 8 months ago
  38. asnaseer
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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

    • 8 months ago
  39. asnaseer
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    |dw:1375035852984:dw|

    • 8 months ago
  40. mathcalculus
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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

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

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

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

    • 8 months ago
  44. asnaseer
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    |dw:1375035980350:dw|

    • 8 months ago
  45. mathcalculus
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    oh okay so i can use 3.. great

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

    • 8 months ago
  47. asnaseer
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    |dw:1375036030210:dw|

    • 8 months ago
  48. asnaseer
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    make sense?

    • 8 months ago
  49. asnaseer
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    |dw:1375036145535:dw|

    • 8 months ago
  50. mathcalculus
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    got it

    • 8 months ago
  51. asnaseer
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    good. now we can examine the lmits

    • 8 months ago
  52. mathcalculus
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    ok :)

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

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

    • 8 months ago
  55. asnaseer
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    |dw:1375036264164:dw|

    • 8 months ago
  56. mathcalculus
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    1 zt 2^-

    • 8 months ago
  57. mathcalculus
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    at*

    • 8 months ago
  58. mathcalculus
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    why is it -1 and not 1?

    • 8 months ago
  59. asnaseer
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    ?

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

    • 8 months ago
  61. asnaseer
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    why is it going to be negative?

    • 8 months ago
  62. mathcalculus
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    but where do i look, in the y?

    • 8 months ago
  63. mathcalculus
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    because it's coming from the left?

    • 8 months ago
  64. asnaseer
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    lets approach the limit in steps

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

    • 8 months ago
  66. asnaseer
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    |dw:1375036393036:dw|

    • 8 months ago
  67. asnaseer
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    |dw:1375036417600:dw|

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

    • 8 months ago
  69. mathcalculus
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    so its 3?

    • 8 months ago
  70. asnaseer
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    perfect! well done! :)

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

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

    • 8 months ago
  73. asnaseer
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    bingo! you're a pro now! :)

    • 8 months ago
  74. mathcalculus
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    2 is undefined?

    • 8 months ago
  75. asnaseer
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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)?

    • 8 months ago
  76. asnaseer
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    yes!

    • 8 months ago
  77. asnaseer
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    I believe you have grokked this now! :)

    • 8 months ago
  78. mathcalculus
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    because, it's not coming from either side.

    • 8 months ago
  79. mathcalculus
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    omg thank you!!!! :))

    • 8 months ago
  80. mathcalculus
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    thank you so much!!

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

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

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

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

    • 8 months ago
  85. mathcalculus
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    do we simply substitute?

    • 8 months ago
  86. mathcalculus
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    @asnaseer

    • 8 months ago
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