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One Sided Limit

  1. Sep 25, 2007 #1
    1. The problem statement, all variables and given/known data

    Evaluate the following limit, if possible:

    lim sq(x² - 10x + 25)
    x->5+ x - 5

    2. Relevant equations



    3. The attempt at a solution

    (x-5)(x-5)
    (x-5)

    lim x - 5
    x->5+

    0?

    x approaches 0? How does the + (one sided) come into play?
     
  2. jcsd
  3. Sep 25, 2007 #2

    Dick

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    I assume the 'sq' means square root. If so can you express the numerator in terms of an absolute value? You'll find the value is quite different if x is a little less than five versus a little larger than five.
     
  4. Sep 25, 2007 #3
    yea i forgot about the square root

    sqr[ (x-5)² ]
    (x-5)

    So the numerator would just be (x-5) after the sq is canceled out.

    Are you saying since x approaches 5 from the right (positive), we only consider the absolute value of x-5? Why would I only apply the absolute value to the numerator?
     
  5. Sep 25, 2007 #4

    Dick

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    The square root of a number is defined to be the positive square root. The numerator is |x-5|.
     
  6. Sep 25, 2007 #5
    Ok, that makes sense.

    ok so it's reduced down to

    |x-5|
    x-5

    i like to make tables:
    x|y
    3|-1
    4|-1
    5|0?
    6|1
    7|1

    I remember my teacher saying something special about 0/0. Does that mean this limit does not exist?

    So the limit as x->5+ does not exist.

    Would it be correct to say the limit as x->5- does not exist?
     
  7. Sep 25, 2007 #6

    Dick

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    Noooo! The limit as x->5+ is the limit of a sequence with x values like 5.1, 5.01, 5.001... The value of the function at the limit doesn't have to be defined for a limit to exist.
     
  8. Sep 25, 2007 #7
    Haha I think I get it now. When we say x->5+ we pick a value to the right of 5 and make it smaller so it approaches 5. So as x approaches 5, the limit is 1.

    The limit of x->5- would be -1

    And the limit of x->5 would be DNE because the limit from the left does not equal the limit from the right.
     
  9. Sep 25, 2007 #8

    Dick

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    Bingo, you've got it.
     
  10. Sep 25, 2007 #9
    Thank you so much
     
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