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Undefined Limits as k-> 0

  1. Apr 30, 2008 #1
    Undefined Limits as k--> 0

    1. The problem statement, all variables and given/known data

    lim(k-->0) [ (-mg)/k + v*e^(kt/m) + (mg)/k*e^(kt/m)]

    ,the end result of this limit is ultimately supposed to be v -gt (or the velocity of an object at any time t neglecting air resistance).

    2. Relevant equations

    This equation comes from the differential equation dv/dt - k/m *v =g

    ,then using integrating factors (the equation itself is a linear ODE) I found:

    v = (-mg)/k + ce^(kt/m)

    ,where c is found by solving for the initial condition v(0)=v0 where

    v0 = (-mg)/k + c(1)

    --> c = v0 + mg/k

    3. The attempt at a solution

    I've spent literally a few hours pouring over this, frustrated as hell that I couldn't solve a simple limit!!

    I tried first taking the natural log of the whole thing, then I tried using all of the rules of exponents separating the e^(x)s. Then I tried working with the differential equation for a while but ultimately was never able to find a way to fully end up with a defined answer - ie. I was unable to completely eliminate x^(-1) or ln(x) prior to taking the limit as x-->0.
  2. jcsd
  3. Apr 30, 2008 #2

    George Jones

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    Rewrite the solution as ( I hope I haven't made a mistake)

    [tex]v \left( t \right) = \exp \left( \frac{k}{m} t \right) v_0 +gm \frac{\exp \left( \frac{k}{m} t \right) - 1}{k},[/tex]

    and then take the limit [itex]k \rightarrow 0[/itex].

    Note that if you set [itex]k = 0[/itex] in the original differential equation, the solution is [itex]v = v_0 + gt[/itex], not [itex]v = v_0 - gt[/itex].
    Last edited: Apr 30, 2008
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