Final Value Theorem Rule Clarification

[chrispypatt]

My homework problem is as follows:
Consider the Laplace transform shown below.
(4s³+15s²+s+30)/(s²+5s+6)

a. What is the value of f(t=0) and f(t=∞)? Use the initial and final value theorems.
b. Find the inverse transform f(t). Use this expression to find f(t=0) and f(t=∞) and compare with the result of part a).I know to find the final value, using FVT, lim_t->inff(t)=lim_s->0sF(s), but I am given the stipulation that the poles must be in the left side of the domain. My book words the definition of IVT and FVT by saying IVT is only valid if f(t) has no impulse functions (the function must be rational) and that for FVT, we must add the rule about the poles. I am confused because everywhere online, including how my professor explained it, is that the impulse rule applies to ONLY IVT and the poles ONLY applies to FVT.

If I went with how my prof taught, Neither theorem can be applied to the problem but if I go by the book's wording, IVT does not apply but FVT shows the final value of f(t) will go to 0.

If someone more knowledgeable with these two theorems could clarify this, that would be helpful! Thanks in advance.

 

[Mark44]

In future posts, please don't delete the homework template -- its use is required.

My homework problem is as follows:
Consider the Laplace transform shown below.
(4s³+15s²+s+30)/(s²+5s+6)

Presumably, the above is F(s); i.e., $\mathcal{L}[f(t)]$.

a. What is the value of f(t=0) and f(t=∞)? Use the initial and final value theorems.
b. Find the inverse transform f(t). Use this expression to find f(t=0) and f(t=∞) and compare with the result of part a).I know to find the final value, using FVT, lim_t->inff(t)=lim_s->0sF(s), but I am given the stipulation that the poles must be in the left side of the domain.

I'm not familiar with either of these theorems, and I don't understand how this stipulation fits in.
What I do know is that $\mathcal{L}[f'(t)] = sF(s) - f(0)$, where F(s) is as above.
For the b) part, I would carry out long division, and for the remainder, use partial fractions to write it as a sum of two fractions.

My book words the definition of IVT and FVT by saying IVT is only valid if f(t) has no impulse functions (the function must be rational) and that for FVT, we must add the rule about the poles. I am confused because everywhere online, including how my professor explained it, is that the impulse rule applies to ONLY IVT and the poles ONLY applies to FVT.

If I went with how my prof taught, Neither theorem can be applied to the problem but if I go by the book's wording, IVT does not apply but FVT shows the final value of f(t) will go to 0.

If someone more knowledgeable with these two theorems could clarify this, that would be helpful! Thanks in advance.

 

[Ray Vickson]

My homework problem is as follows:
Consider the Laplace transform shown below.
(4s³+15s²+s+30)/(s²+5s+6)

a. What is the value of f(t=0) and f(t=∞)? Use the initial and final value theorems.
b. Find the inverse transform f(t). Use this expression to find f(t=0) and f(t=∞) and compare with the result of part a).I know to find the final value, using FVT, lim_t->inff(t)=lim_s->0sF(s), but I am given the stipulation that the poles must be in the left side of the domain. My book words the definition of IVT and FVT by saying IVT is only valid if f(t) has no impulse functions (the function must be rational) and that for FVT, we must add the rule about the poles. I am confused because everywhere online, including how my professor explained it, is that the impulse rule applies to ONLY IVT and the poles ONLY applies to FVT.

If I went with how my prof taught, Neither theorem can be applied to the problem but if I go by the book's wording, IVT does not apply but FVT shows the final value of f(t) will go to 0.

If someone more knowledgeable with these two theorems could clarify this, that would be helpful! Thanks in advance.

A lot of the on-line literature on these matters is confusing, and sometimes even self-contradictory (proving the result in one section, then showing a couner-example in another section). For the FVT, a nice article that sets it out properly and clearly is
http://www.me.umn.edu/courses/me3281/notes/TransformSolutionsToLTISystems_Part4.pdf .