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Math Help - doubt

  1. #1
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    doubt

    if i have a function defined like this:
    f(x)=(1+x)^(2/x) for x>0
    k for x=0
    at zero 0 the function is continuous for which value of k? but, to be continuous the limit must exist and be equal to the value of the function at that point; i might say e^2 but actually the left limit does not exist...that is what makes me perplex..
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  2. #2
    TD!
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    Your answer is correct. Why do you think the left limit doesn't exist?
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  3. #3
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    because the function is defined only for x>=0
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  4. #4
    TD!
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    Oh I didn't read that.

    What's your thought on this: is f(x) = sqrt(x) continuous at x = 0?
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  5. #5
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    well the value exists at x=0 so why should it be discontinuos there?i think it is contiuous..
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  6. #6
    TD!
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    Being defined at x = a isn't a sufficient condition for being continous there, but you are correct: sqrt(x) is continuous at x = 0.
    However, the left limit for x approaching 0 of sqrt(x) is meaningless, since sqrt(x) is only defined for x > 0.

    Do you see where I'm going?
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  7. #7
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    that we have to take into consideration the domain too? because the very question of concerning about the left limit is wrong as it is out o the domain, of what interests us?
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  8. #8
    TD!
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    Since f isn't defined for x < 0, it is meaningless to look at the left limit for x approaching 0.
    In this case, you are only to consider the right limit since f is only defined for x > 0.
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  9. #9
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    ok, so the same for the previous function; i don't have to concern myself about a thing, a part that is not contemplated by the function; k=e^2 then is right, isn' t it?
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  10. #10
    TD!
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    That's the correct answer indeed!
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  11. #11
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    you're really kind!!!you made me reason about that!thank you!
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  12. #12
    TD!
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    You're welcome
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  13. #13
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    The general result of what TD! said is like this.

    A function is said to be continous on a closed interval [a,b] is it is continous on (a,b) and the limit from the left at a is f(a) and the limit from the right at b is f(b).
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