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lim k->0 (e^kt-a)/k = at

It seems strange to me, and I have no idea how to derive it. Here's how I got it. Starting with:

dv/dt = a+kv

Seperate variables and integrate to get:

dv/(a+kv) = dt

(1/k)*ln(a+kv) = t+c (I'll set c=0 from here on)

a+kv = e^(kt)

v = (e^kt-a)/k

which is the left side of the limit. But as k goes to 0 in the original equation:

dv/dt = a

v = at

Is this right? If so, is there a better way to derive it? If not, where did I go wrong? (I know there are multiple divide by zeros, but I don't think that's the problem because k varies continuously, and the original curve could get closer and closer to a straight line as k is smaller and smaller but still greater than 0.)

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# Is this a valid limit?

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