Estimate Limits Numerically

Discussion in 'Calculus' started by nycmathguy, Oct 7, 2021.

  1. nycmathguy

    nycmathguy

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    Exercises 1.2
    6 & 8

    20211006_201954.jpg

    Question 6

    The purpose of this table is to see what happens as we get closer and closer to 3. I say x cannot be the number in the middle of the table or 3. I will not evaluate this function at the given x-values here. I just want to know if my reasoning is correct.

    Note: The graph tells a different story. I think x tends to -3 not 3. You say?

    20211006_232311.jpg

    Question 8

    The purpose of this table is to see what happens as we get closer and closer to 3. I say x cannot be the number in the middle of the table or 3. I will not evaluate this function at the given x-values here. I just want to know if my reasoning is correct.

    Note: The graph tells a different story. Does x really tend to 3?

    20211006_233426.jpg
     
    Last edited: Oct 7, 2021
    nycmathguy, Oct 7, 2021
    #1
  2. nycmathguy

    MathLover1

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    reasoning is correct

    6.
    if you calculate this way, as x->3 limit will be undefined
    table
    x ........... 2.9 ........... 2.99 ........... 2.999 ........... 3 ........... 3.001 ........... 3.01 ........... 3.1
    f(x) ........ 0.17 ........ 0.17 ........ 0.167 ........ (undefined) ........ 0.167 ........ 0.167 ........ 0.16

    but, if you simplify function you got y=1/(x+3),
    table
    x ........... 2.9 ... ........ 2.99 ........... 2.999 ........... 3 ........... 3.001 ........... 3.01 ........... 3.1
    f(x) ........ 0.17 ........ 0.167........ 0.16669........ (1/6) ........ 0.16666 ........ 0.166 ........ 0.164

    then as x->3 limit will be 1/6

    [​IMG]
     
    Last edited: Oct 7, 2021
    MathLover1, Oct 7, 2021
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  3. nycmathguy

    MathLover1

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    8.

    [​IMG]

    [​IMG]

    [​IMG]

    [​IMG]

    so [​IMG]

    when you do the table, you see as x->3 that f(x)=-1/16

    [​IMG]
     
    MathLover1, Oct 7, 2021
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  4. nycmathguy

    nycmathguy

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    Last edited: Oct 7, 2021
    nycmathguy, Oct 7, 2021
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  5. nycmathguy

    nycmathguy

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    Using the table is very tedious. This exercise is basically to show students how to find the limit as x tends to a certain value without actually getting there.
     
    nycmathguy, Oct 7, 2021
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  6. nycmathguy

    MathLover1

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    When it has a minus sign at the top it means that you must find the limit when the desired x comes from the left, meaning that you must find which number the limit goes as x becomes more closer to 3 from the left.
     
    MathLover1, Oct 7, 2021
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  7. nycmathguy

    nycmathguy

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    Yes, I figured that out by reading your reply twice.
     
    nycmathguy, Oct 7, 2021
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