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Math Help - True or false on certain limit "theory" questions

  1. #1
    s3a
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    True or false on certain limit "theory" questions

    The following are the questions I am having trouble understanding. (The answers in the boxes are the correct answers)

    "(1 pt) Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. A statement is true only if it is true for all possibilities. You must get all of the answers correct to receive credit.

    1. If is continuous at , then is differentiable at
    2.
    3. If and , then does not exist
    4. If and , then does not exist
    5. "

    I have the answers but I was hoping someone could at least briefly explain why each one is true or false.

    Any help would be greatly appreciated!
    Thanks in advance!
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  2. #2
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    Put some spaces between those questions - they are bleeding into each other and I have crappy eyes.

    Are you having trouble answering all of the problems?
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  3. #3
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    Quote Originally Posted by s3a View Post
    The following are the questions I am having trouble understanding. (The answers in the boxes are the correct answers)

    "(1 pt) Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. A statement is true only if it is true for all possibilities. You must get all of the answers correct to receive credit.

    1. If is continuous at , then is differentiable at
    2.
    3. If and , then does not exist
    4. If and , then does not exist
    5. "

    I have the answers but I was hoping someone could at least briefly explain why each one is true or false.

    Any help would be greatly appreciated!
    Thanks in advance!

    Next time do "preview" your answer to avoid cumbersome-looking posts as the above...

    1) False: counterexample f(x)=|x| at zero

    2) True since both limits exist and are finite and the denominator's limit isn't non-zero

    3) False: counterexample f(x)=g(x)=x-4 .

    4) True, IFFFFF by "doesn't exist" you people mean the limit doesn't exists FINITELY. For example, \lim_{x\to 4}\frac{x+1}{(x-4)^2}=\infty exists in the generalized sense. It all depends on your definitions.

    5) Obviously false since then you'd get \frac{0}{0} ...The actual limit is 7/5 .

    Tonio
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  4. #4
    s3a
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    Sorry for not previewing. Also thanks for the answer, but could you just confirm the following (in my own words) for me please?:

    #3. the limit = 1 therefore, it does exist? (because something over same thing = 1)
    #4. the limit = 5/0 therefore, it does not exist?
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    Quote Originally Posted by s3a View Post
    Sorry for not previewing. Also thanks for the answer, but could you just confirm the following (in my own words) for me please?:

    #3. the limit = 1 therefore, it does exist? (because something over same thing = 1)

    You don't have "something over the same" here, but two things that converge to the same AND DIFFERENT from zero, thus the limit exists and it's possible to evaluate it the way it's done in the question.


    #4. the limit = 5/0 therefore, it does not exist?

    You can't write that since \frac{anything}{0} isn't defined in mathematics. What you can say is that while the numerator approaches a finite number as x\xrightarrow [x\to 1] {}1, the denominator approaches \infty and then you cannot try to evaluate the limit the way they're showing in the question.

    Tonio
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