I'm looking to solve the limit of the following error function as s goes to 0, but I'm failing on factoring things out. My calculator (TI-89) gives me a nice form I can use, but I cannot manipulate things quite right. Can anyone point out what I'm doing wrong/missing?
The function starts as $$sE(s)=\frac{s}{s^2} \left[1-\frac{K(K_1s + K_2)}{Ts^2 + (K K_1 +1)s + K K_2}\right]$$ I multiplied so as to combine the two fractions: $$\frac{1}{s}\cdot\frac{Ts+ KK_1 + 1 + \frac{KK_2}{s}}{Ts+ KK_1 + 1 + \frac{KK_2}{s}}$$ And I'm left with $$\frac{Ts+ KK_1 + 1 + \frac{KK_2}{s} - KK_1s - KK_2}{Ts^2 + (K K_1 +1)s + K K_2}$$
Unfortunately, I can't figure how to simplify out the numerator.
The calculator states the answer is $$\frac{Ts+1}{Ts^2+(K K_1 +1)s+K K_2}$$ I can deal with that, as the limit will simply be $\frac{1}{KK_2}$. But what are the steps in between? I don't know how to get rid of the $KK_x$ terms in the numerator as I need to.
Thanks in advance.
I believe your mistake is somewhere in how you combined the fractions initially. Here's what the simplification should look like: \begin{align*} sE(s)&=\frac{s}{s^2} \left[1-\frac{K(K_1s + K_2)}{Ts^2 + (K K_1 +1)s + K K_2}\right]\\ &=\frac{1}{s} - \frac{K(K_1s + K_2)}{s(Ts^2 + (K K_1 +1)s + K K_2)}\\ &=\frac{Ts^2 + (K K_1 +1)s + K K_2-K(K_1s + K_2)}{s(Ts^2 + (K K_1 +1)s + K K_2)}\\ &=\frac{Ts^{2}+s}{s(Ts^2 + (K K_1 +1)s + K K_2)}\\ &=\frac{Ts+1}{Ts^2 + (K K_1 +1)s + K K_2} \end{align*}