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Math Help - How do I get(prove/solve) these limits? "lim x->+infinity (x)^(2/ln(1+x^3))" ...?

  1. #1
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    Question How do I get(prove/solve) these limits? "lim x->+infinity (x)^(2/ln(1+x^3))" ...?

    Hello, i've got some questions i really don't know how to prove/solve, and thats limits of infinity involving fraction as exponents.(lim x->+infinity (x)^(2/ln(1+x^3)) ) .

    here are some questions that puzzles me on:

    pi = the "pi" sign
    sqrt. of = sqrt. of a number
    infinity = infinity sign

    1.) lim x->+infinity (x)^(2/ln(1+x^3))
    2.) lim x->0^+ (cos sqrt of x)^(1/x)
    3.) limx->1-0 ((2/pi)(arcsin(x)))^(1/acrcos(x))
    4.) lim x->0^+ (x)^((1)/(1+ln sqrt. of x))
    5.) lim x->+infinity (2+e^x)^(-1/x)
    6.) lim x->0 (1+3x)^(2/sin(x))
    7.) lim x->0 ((a^x+b^x+c^x)/(3))^(1/x)


    well, I've already encounter and solve some limits but not related on these parts(fractions as exponents)
    tried reading my book but didn't show some examples on how to solve it.

    If you know how to solve it, please let me know.
    or if you've got some examples,then show it to me.
    And
    thanks for your time!
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  2. #2
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    Quote Originally Posted by tadomore View Post
    Hello, i've got some questions i really don't know how to prove/solve, and thats limits of infinity involving fraction as exponents.(lim x->+infinity (x)^(2/ln(1+x^3)) ) .

    here are some questions that puzzles me on:

    pi = the "pi" sign
    sqrt. of = sqrt. of a number
    infinity = infinity sign

    1.) lim x->+infinity (x)^(2/ln(1+x^3))
    2.) lim x->0^+ (cos sqrt of x)^(1/x)
    3.) limx->1-0 ((2/pi)(arcsin(x)))^(1/acrcos(x))
    4.) lim x->0^+ (x)^((1)/(1+ln sqrt. of x))
    5.) lim x->+infinity (2+e^x)^(-1/x)
    6.) lim x->0 (1+3x)^(2/sin(x))
    7.) lim x->0 ((a^x+b^x+c^x)/(3))^(1/x)


    well, I've already encounter and solve some limits but not related on these parts(fractions as exponents)
    tried reading my book but didn't show some examples on how to solve it.

    If you know how to solve it, please let me know.
    or if you've got some examples,then show it to me.
    And
    thanks for your time!


    Have you studied L'Hospital rule?

    \displaystyle{\lim\limits_{x\rightarrow\infty}\fra  c{\ln x}{\ln(1+x^3)}=\lim\limits_{x\rightarrow\infty}\fr  ac{x^3+1}{3x^3}=\frac{1}{3}} , applying L'H rule, so

    \displaystyle{x^{\frac{2}{\ln(1+x^3)}}=e^{2\frac{\  ln x}{\ln(1+x^3)}}\xrightarrow [x\to\infty]{}e^{2/3}} .

    Without L'Hospital I don't know how to evaluate the above limit, and most of the other ones you wrote.

    Tonio
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    Quote Originally Posted by tonio View Post
    Have you studied L'Hospital rule?

    \displaystyle{\lim\limits_{x\rightarrow\infty}\fra  c{\ln x}{\ln(1+x^3)}=\lim\limits_{x\rightarrow\infty}\fr  ac{x^3+1}{3x^3}=\frac{1}{3}} , applying L'H rule, so

    \displaystyle{x^{\frac{2}{\ln(1+x^3)}}=e^{2\frac{\  ln x}{\ln(1+x^3)}}\xrightarrow [x\to\infty]{}e^{2/3}} .

    Without L'Hospital I don't know how to evaluate the above limit, and most of the other ones you wrote.

    Tonio
    we have discuss it last time but the teacher didn't give at least one sample on solving it(and i didn't quite understand it yet).
    anyway mind to check if my answers for no.2,4, and 5 are correct
    in no. 2 I get an answer of -0.5 ; no.4 e^2 ; no.5 -1
    while i'm still tryingto solve 3,6,7
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  4. #4
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    Quote Originally Posted by tadomore View Post
    we have discuss it last time but the teacher didn't give at least one sample on solving it(and i didn't quite understand it yet).
    anyway mind to check if my answers for no.2,4, and 5 are correct
    in no. 2 I get an answer of -0.5 ; no.4 e^2 ; no.5 -1
    while i'm still tryingto solve 3,6,7


    The answer in 2 is e^{-1/2} , not -1/2 , 4 is correct and you don't even need limits for that but only simple

    properties of logarithms, and the answer in 5 is e^{-1} , not -1...you seem to be forgetting the exponential function.

    Tonio
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