How can I find the constants A and B if <munder> <mo movablelimits="true" form="prefix">lim

George Bray

George Bray

Answered question

2022-06-06

How can I find the constants A and B if
lim x ( x 2 + x A x B ) = 0 ?

Answer & Explanation

Judovh0

Judovh0

Beginner2022-06-07Added 16 answers

You're trying to find the asymptote of x 2 + x at infinity. The standard method is to say that if
lim x x 2 + x A x B = 0
then surely also
lim x x 2 + x A x B x = 0
which implies
A = lim x x 2 + x x = lim x 1 + 1 x = 1
So A = 1 and now you can substitute this back into the original equation to find B.
Summer Bradford

Summer Bradford

Beginner2022-06-08Added 11 answers

( ( x + 1 / 2 ) 2 1 / 4 ) 1 / 2 A x B ;
Set y := x + 1 / 2 , and consider y
( y 2 1 / 4 ) 1 / 2 A ( y 1 / 2 ) B ;
The asymptote of ( y 2 1 / 4 ) 1 / 2 is y.
Collecting corresponding terms in the asymptotic expression:
y ( 1 A ) + ( ( 1 / 2 ) A B ) = 0 we get
A = 1 and B = 1 / 2.

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