Finding A Limit Graphically...2

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

  1. nycmathguy

    nycmathguy

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    Exercises 1.2
    25 - 28

    20211006_201853.jpg

    25)

    The limit from the left does not equal the limit from the right. Thus, no limit.

    26)

    The limit from the left is negative infinity. The limit from the right is positive infinity. The limit from the left does not equal the limit from the right. So, no limit.

    27)

    This is an oscillating function. The limit does not exist.

    28)

    The limit from the left is positive infinity.
    The limit from the right is negative infinity.
    The limit from the left does not equal the limit from the right. So, the limit does not exist.

    You say?
     
    nycmathguy, Oct 7, 2021
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  2. nycmathguy

    MathLover1

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    25)
    there are holes in (2,1) and (2,-1)
    as x comes closer and closer to 2 from the left side, limit is getting closer and closer to -1 (but never gets equal to -1) even we write that [​IMG]

    as x comes closer and closer to 2 from the right side, limit is getting closer and closer to 1

    [​IMG]

    26. correct

    27. might exist but indeterminate
    [​IMG]

    28. correct

    also,
    [​IMG]
     
    MathLover1, Oct 7, 2021
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  3. nycmathguy

    nycmathguy

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    For 27 you said the limit "might exist but indeterminate." How can we know for sure?
    Does it involve advanced limit studies?

    For 25, can you show me how to find the limit algebraically? What if I let x = 2?

    |2 - 2|/(2 - 2) = |0|/0 = 0/0 = indeterminate. Yes?
     
    nycmathguy, Oct 7, 2021
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  4. nycmathguy

    MathLover1

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    yes, or you can say undefined
     
    MathLover1, Oct 7, 2021
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  5. nycmathguy

    nycmathguy

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    Do you mean to show 27 algebraically involves further limit studies?
    We can say that 25 is undefined? How does
    L' Hopital's Rule apply when we get 0/0?
     
    nycmathguy, Oct 7, 2021
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  6. nycmathguy

    MathLover1

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    just for 27

    25. as x comes closer and closer to 2 (but never get 2) from the right side, limit is getting closer and closer to 1
     
    MathLover1, Oct 7, 2021
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  7. nycmathguy

    nycmathguy

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    In terms of limits, all oscillating functions are undefined. Yes?
     
    nycmathguy, Oct 7, 2021
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  8. nycmathguy

    MathLover1

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    some oscillating functions have no limit , some do have a limit

    Some trigonometric functions start to oscillate so radically between two y-values as x approaches a specific value the limit is undefined. The graph of f(x) = cos(1/x) is shown in this second graph appearing here:.

    upload_2021-10-7_16-53-49.png
    We can see from our second graph that as x approaches 0 from either side the graph becomes like a compressed accordion shape that is oscillating in an extreme fashion, which is an undefined limit in calculus! This is called an infinite oscillator.

    or, f(x)=sin(1/x)
    upload_2021-10-7_16-52-42.png

    This function doesn't have a limit as x→0 since it just oscillates more and more wildly between -1 and 1.
    so, this function undefined limit

    or, f(x) = 1/x^2
    upload_2021-10-7_16-51-43.png
    This function exists for all nonzero values of x, and doesn't have a limit as x→0
     
    Last edited: Oct 7, 2021
    MathLover1, Oct 7, 2021
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  9. nycmathguy

    nycmathguy

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    Thank you so much. I'm so glad too have met you. I have learned do much math thanks to you. This is just the beginning of a long, mathematical journey.
     
    nycmathguy, Oct 8, 2021
    #9
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