Differential equation involving falling object

[MissMCHP]

Homework Statement

A body of mass m is projected vertically upward with an initial velocity v₀ in a medium offering a resistance k|v|, where k is a constant. Assume that the gravitational attraction of the Earth is constant.
a) Find the velocity v(t) of the body at any time.
b) Using the result of part a) to calculate the limit of v(t) as k->0

Homework Equations

The Attempt at a Solution

So I managed part a) with a solution of

v(t) = -mg/k + (v₀+mg/k)e^-kt/m,

which after checking and re-checking, I am fairly confident in. The problem is that when I try to find the limit as k->0, all I could come up with is v(t)=v₀. This is obviously wrong since the as k->0, air resistance become negligible, and the answer should be the all too familiar v(t) = v₀-gt. I tried to graph the function, and I found that after picking out all the useless part, it boils down to

lim _k->t m/k(1-e^-kt/m) = t!

Well. It does make sense, but I could not for the life of me figure out how to derive it mathematically. I am stuck, and any help would be greatly appreciated!

 

[Dick]

Good deductive work on figuring out what the limit MUST be, you meant k->0, right? Now use l'Hopital's rule. It's a limit of the form 0/0. The limit is t!

 

[MissMCHP]

Yes! I did mean k->0. Thank you! I never would have thought of L'Hopital's rule (even though I should)