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KCC's quizzes AQQ255 about a limit at infinite

And here is a new quiz coming!

1. Info: the frequent winners should have received some award and certificate. Please warn me if you think you have been forgotten!

2. Quote of the week: "The elevator to success is out of order. You'll have to use the stairs... one step at a time." - Joe Girard.

3. New quiz of the week: AQQ255 about a limit to infinite

Good luck!

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[edited by: emassa at 3:52 PM (GMT -4) on 21 Mar 2024]
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  • 1. This is the constant e (2.71828...)

    2. If circular proofs are allowed and we know of the constant e.

    Applying limits (1+1/infinite)^(infinite). This means if we rewrite the base as f(x) = (1+1/infinite) and g(x) as the exponent that approaches infinity, we get an indeterminate form as f(x) from the right approaches 1 while the exponent g(x) forces the base to go infinite.

    Let h(x) = (1+1/x)^x

    Ln h(x) = x * ln(1+1/x)

    Reform the right hand side and drop x to the denominator as 1/x.

    Ln h(x) = ln(1+1/x)/(1/x)

    Apply limits as x approaches infinity and we get 0/0 which is indeterminate.

    We can use L'hospital's rule such that:

    m(x) = ln(1+1/x)

    n(x) = 1/x

    m'(x) = -(1/(x^2+x))

    n'(x) = -1/x^2


    Ln h(x) = (1/(1+1/x))

    Apply the limits as x approaches infinity and we get...

    Ln h(x) = 1

    Since we know ln is log of base e, we can revert it as h(x) = e.

    Not a good proof but I'll reply if I can think of a better one.

  • Yes! You applied the right approach to prove it in a very formal way! Congratulation  ! The Hospital method saying Lim[(Fx)/G(x)] = Lim[(F’x)/G’x)] gives quickly the solution.

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