# Initial and final value theorems

1. Dec 4, 2013

### bl4ke360

1. The problem statement, all variables and given/known data
Find f(t) for the function F(s)=(10s^2+85s+95)/(s^2+6s+5) and apply the initial and final value theorems to each transform pair

2. Relevant equations
Initial value theorem: f(0)=lim s->∞ s(F(s))
Final value theorem: f(∞) = lim s->0 s(F(s))

3. The attempt at a solution
After dividing due to improper fraction:
F(s)= 10 + (25s+45)/(s^2+6s+5)

F(s)= 10+5/(s+1)+20/(s+5)
f(t)= 10δ(t)+[5e^(-t)+20e^(-5t)]u(t)

Where I'm confused is how I would apply the value theorems since there's an impulse function. When my professor did a similar problem and applied the theorems, I couldn't follow what she did, but the answer solution to this problem says the value theorems can't be applied to the function because the function is improper and the corresponding f(t) function contains an impulse.
How was my professor able to do it if it supposedly can't be done? Can someone please clarify this for me?

2. Dec 5, 2013

### ShayanJ

Which transform are you talking about? Fourier,Laplace or what?

3. Dec 5, 2013

Laplace

4. Dec 5, 2013

### ShayanJ

Dirac delta is frequently explained as being zero everywhere except at one point which becomes infinite there.So $\lim_{t\rightarrow 0}\delta(t)=0$because although t is very near to 0,but it isn't equal to 0!

EDIT:Looks like I'm wrong...because $\lim_{s\rightarrow\infty}sF(s)=\infty$!
So it seems we should have $\lim_{t\rightarrow 0}\delta(t)=\infty$!!!

5. Dec 5, 2013

### bl4ke360

How did you get infinity for lim s → ∞ sF(s)? This says it's 25:

http://www.wolframalpha.com/input/?i=lim+as+x-%3Einfinity+%2825x^2%2B45x%29%2F%28x^2%2B6x%2B5%29

6. Dec 5, 2013

### ShayanJ

You forgot the 10s part!

7. Dec 5, 2013

### bl4ke360

Are you sure the limit applies to the 10 as well? When my professor did this type of problem she only applied the limit to the fractional part. I'm so confused because my professor, the solutions manual, and you are all saying different things so I have no idea what is correct.

8. Dec 5, 2013

### ShayanJ

Its simple.The theorem urges us to find $\lim_{s \rightarrow \infty} sF(s)$ and whether you like it or not,10s is part of sF(s)!
Also you can tell us what your solution manual and your teacher say,not only the results,but also the explanations!

9. Dec 5, 2013

### bl4ke360

Well I won't have time for that, my final is in 10 hours lol. This is what the solution manual says:

10. Dec 5, 2013

### ShayanJ

Well,I think that's right.Because I was right about the limit of dirac delta being zero as the argument goes to zero.www.Wolframalpha.com confirms that! and because sF(s) becomes infinite as s goes to infinity,you can't apply the theorem.