• Support PF! Buy your school textbooks, materials and every day products Here!

Undefined Limits as k-> 0

  • Thread starter pr0me7heu2
  • Start date
  • #1
14
2
Undefined Limits as k--> 0

Homework Statement



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).



Homework 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

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.
 

Answers and Replies

  • #2
George Jones
Staff Emeritus
Science Advisor
Gold Member
7,261
790
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:

Related Threads for: Undefined Limits as k-> 0

  • Last Post
Replies
14
Views
2K
  • Last Post
Replies
7
Views
2K
Replies
3
Views
5K
Replies
0
Views
959
  • Last Post
Replies
9
Views
2K
  • Last Post
Replies
6
Views
973
Replies
13
Views
2K
Top