Arectemieryf0

Answered

2022-07-20

Not seeing steps between factoring fractions

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

$sE(s)=\frac{s}{{s}^{2}}[1-\frac{K({K}_{1}s+{K}_{2})}{T{s}^{2}+(K{K}_{1}+1)s+K{K}_{2}}]$

I multiplied so as to combine the two fractions:

$\frac{1}{s}\cdot \frac{Ts+K{K}_{1}+1+\frac{K{K}_{2}}{s}}{Ts+K{K}_{1}+1+\frac{K{K}_{2}}{s}}$

And I'm left with

$\frac{Ts+K{K}_{1}+1+\frac{K{K}_{2}}{s}-K{K}_{1}s-K{K}_{2}}{T{s}^{2}+(K{K}_{1}+1)s+K{K}_{2}}$

Unfortunately, I can't figure how to simplify out the numerator.

The calculator states the answer is

$\frac{Ts+1}{T{s}^{2}+(K{K}_{1}+1)s+K{K}_{2}}$

I can deal with that, as the limit will simply be $\frac{1}{K{K}_{2}}$. But what are the steps in between? I don't know how to get rid of the $K{K}_{x}$ terms in the numerator as I need to.

Thanks in advance.

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

$sE(s)=\frac{s}{{s}^{2}}[1-\frac{K({K}_{1}s+{K}_{2})}{T{s}^{2}+(K{K}_{1}+1)s+K{K}_{2}}]$

I multiplied so as to combine the two fractions:

$\frac{1}{s}\cdot \frac{Ts+K{K}_{1}+1+\frac{K{K}_{2}}{s}}{Ts+K{K}_{1}+1+\frac{K{K}_{2}}{s}}$

And I'm left with

$\frac{Ts+K{K}_{1}+1+\frac{K{K}_{2}}{s}-K{K}_{1}s-K{K}_{2}}{T{s}^{2}+(K{K}_{1}+1)s+K{K}_{2}}$

Unfortunately, I can't figure how to simplify out the numerator.

The calculator states the answer is

$\frac{Ts+1}{T{s}^{2}+(K{K}_{1}+1)s+K{K}_{2}}$

I can deal with that, as the limit will simply be $\frac{1}{K{K}_{2}}$. But what are the steps in between? I don't know how to get rid of the $K{K}_{x}$ terms in the numerator as I need to.

Thanks in advance.

Answer & Explanation

lelapem

Expert

2022-07-21Added 12 answers

I believe your mistake is somewhere in how you combined the fractions initially. Here's what the simplification should look like:

$\begin{array}{rl}sE(s)& =\frac{s}{{s}^{2}}[1-\frac{K({K}_{1}s+{K}_{2})}{T{s}^{2}+(K{K}_{1}+1)s+K{K}_{2}}]\\ & =\frac{1}{s}-\frac{K({K}_{1}s+{K}_{2})}{s(T{s}^{2}+(K{K}_{1}+1)s+K{K}_{2})}\\ & =\frac{T{s}^{2}+(K{K}_{1}+1)s+K{K}_{2}-K({K}_{1}s+{K}_{2})}{s(T{s}^{2}+(K{K}_{1}+1)s+K{K}_{2})}\\ & =\frac{T{s}^{2}+s}{s(T{s}^{2}+(K{K}_{1}+1)s+K{K}_{2})}\\ & =\frac{Ts+1}{T{s}^{2}+(K{K}_{1}+1)s+K{K}_{2}}\end{array}$

$\begin{array}{rl}sE(s)& =\frac{s}{{s}^{2}}[1-\frac{K({K}_{1}s+{K}_{2})}{T{s}^{2}+(K{K}_{1}+1)s+K{K}_{2}}]\\ & =\frac{1}{s}-\frac{K({K}_{1}s+{K}_{2})}{s(T{s}^{2}+(K{K}_{1}+1)s+K{K}_{2})}\\ & =\frac{T{s}^{2}+(K{K}_{1}+1)s+K{K}_{2}-K({K}_{1}s+{K}_{2})}{s(T{s}^{2}+(K{K}_{1}+1)s+K{K}_{2})}\\ & =\frac{T{s}^{2}+s}{s(T{s}^{2}+(K{K}_{1}+1)s+K{K}_{2})}\\ & =\frac{Ts+1}{T{s}^{2}+(K{K}_{1}+1)s+K{K}_{2}}\end{array}$

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