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Thread: as time tends to infinity basic confusion!!!

  1. #1
    Member i_zz_y_ill's Avatar
    Mar 2008

    as time tends to infinity basic confusion!!!

    I dont think this answer bares any relevance to he situaton, M is mass t is time and k = 25root2

    have condition M = kt/(1+t^2)^0.5

    the answer as t tends to infinty is that M tends to k.

    I have tried cancelling t's and this doesnt for me but i can only come to the conclusion that its alot bigger or smaller than that,, whats the relevance of 1 aswell.
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  2. #2
    Super Member wingless's Avatar
    Dec 2007
    The question is asking for $\displaystyle \lim_{t\to\infty}\frac{k t}{\sqrt{1+t^2}}$.

    $\displaystyle \lim_{t\to\infty}\frac{k \sqrt{t^2}}{\sqrt{1+t^2}}$

    $\displaystyle \lim_{t\to\infty}k\sqrt{\frac{t^2}{t^2+1}}$

    $\displaystyle k\cdot\lim_{t\to\infty}\underbrace{\sqrt{\frac{t^2 }{t^2+1}}}_{\text{This limit is 1}}$

    $\displaystyle k\cdot 1 = \boxed{~k~}$

    There are lots of methods to find $\displaystyle \lim_{t\to\infty}\sqrt{\frac{t^2}{t^2+1}}$. You can use comparison of degrees and coefficients, L'hopital's rule, simplifying the limit etc.

    Let $\displaystyle I = \lim_{t\to\infty}\sqrt{\frac{t^2}{t^2+1}}$.

    $\displaystyle I\cdot I = \lim_{t\to\infty}\sqrt{\frac{t^2}{t^2+1}}\cdot\lim _{t\to\infty}\sqrt{\frac{t^2}{t^2+1}}$

    $\displaystyle I^2 = \lim_{t\to\infty}\sqrt{\frac{t^2}{t^2+1}}\cdot\sqr t{\frac{t^2}{t^2+1}}$

    $\displaystyle I^2 = \lim_{t\to\infty}\frac{t^2}{t^2+1}$

    In $\displaystyle t^2 + 1$, $\displaystyle t^2$ will grow much faster than 1 and will make 1 meaningless. So we can cross it out:

    $\displaystyle \lim_{t\to\infty}\frac{t^2}{t^2+\not 1} = \frac{t^2}{t^2} = 1$

    So $\displaystyle I^2 = 1$ and $\displaystyle I=1$.
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