I am trying to solve this problem : <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-s

dikcijom2k

dikcijom2k

Answered question

2022-07-07

I am trying to solve this problem :
P = lim x 0 p H + ( 1 p H ) ( 1 x ) t p H + ( 1 p H ) ( 1 x ) t x p L + ( 1 p L ) ( 1 x ) t p L + ( 1 p L ) ( 1 x ) t x x

Answer & Explanation

iskakanjulc

iskakanjulc

Beginner2022-07-08Added 18 answers

Let a = p H and b = p L ; so the numerator of the expression is
A = a + ( 1 a ) ( 1 x ) t a + ( 1 a ) ( 1 x ) t x b + ( 1 b ) ( 1 x ) t b + ( 1 b ) ( 1 x ) t x
Using Taylor expansion around x = 0
( 1 x ) k = n = 0 ( 1 ) n ( k n ) x n
Use it for a very few terms and continue with the long division to obtain
a + ( 1 a ) ( 1 x ) t a + ( 1 a ) ( 1 x ) t x = 1 + ( a 1 ) x 2 1 2 ( a 1 ) ( 2 a t 1 ) x 3 + O ( x 4 )
that is to say
A = ( a b ) x 2 + O ( x 3 ) P = lim x 0 A x = lim x 0 [ ( a b ) x ] = 0
that is to say
A = ( a b ) x 2 + O ( x 3 ) P = lim x 0 A x = lim x 0 [ ( a b ) x ] = 0
which is what you found using L'Hospital.

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