Not seeing steps between factoring fractionsI'm looking to solve the limit of the following error...

Arectemieryf0

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
s E ( s ) = s s 2 [ 1 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:
1 s T s + K K 1 + 1 + K K 2 s T s + K K 1 + 1 + K K 2 s
And I'm left with
T s + K K 1 + 1 + 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
T s + 1 T s 2 + ( K K 1 + 1 ) s + K K 2
I can deal with that, as the limit will simply be 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

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:
s E ( s ) = s s 2 [ 1 K ( K 1 s + K 2 ) T s 2 + ( K K 1 + 1 ) s + K K 2 ] = 1 s K ( K 1 s + K 2 ) s ( T s 2 + ( K K 1 + 1 ) s + K K 2 ) = 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 ) = T s 2 + s s ( T s 2 + ( K K 1 + 1 ) s + K K 2 ) = T s + 1 T s 2 + ( K K 1 + 1 ) s + K K 2

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