# Hard limit problem.

1. Sep 12, 2009

### Eric_meyers

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

"Demonstrate that in the limit (nu)---> 2w0 the solution to the damped harmonic oscillator equation becomes x(t) = (x0 + [v0 +(nu/2)x0]t)e^(-(nu)t/2)

2. Relevant equations

Solution to damped harmonic oscillator equation; x(t) = a*e^(-(nu)t/2)cos(w1t - (theta))

Where a = [x0^2 + (v0 + (nu)x0/2)^2/w1^2]1/2 and

(theta) = tan^-1((v0 + (nu)x0/2)/w1x0)

and w1^2 = w0^2 - (nu)^2/4

3. The attempt at a solution

So I plugged (nu) ---> 2w0 into my equations and got ...

x0^2 + [(v0 + w0x0)^2)/(w0^2 - w0^2)]^1/2 * e^(-w0t) * cos (-tan^-1 (( v0 + w0x0)/((w0^2-w0^2)^1/2*x0))

and I note that the divide by 0 in my tan inverse goes to infinity which equals pi/2 and the cos of pi/2 is 0

but then before my e term I have (w0^2 - w0^2) on the denominator which means that's going to infinity... so I have one part going to 0 and one part going to infinity...

and I don't know how to deal with this.

Last edited: Sep 12, 2009