goldch6 - Contents VI Asymptotic Behaviour of Rational...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Contents VI Asymptotic Behaviour of Rational Functions 1 VI.A Rational Functions as x → ±∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 VI.B Rational Functions Near Zeros of the Denominator; Poles . . . . . . . . . . . . . . . . . . . 7 VI.C Graph Versus Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
VI Asymptotic Behaviour of Rational Functions VI.1 VI Asymptotic Behaviour of Rational Functions VI.A Rational Functions as x → ±∞ There exists a positive real constant R such that all roots of Q ( x ) are in [ - R,R ] and so f ( x ) = P ( x ) Q ( x ) is defined whenever x < - R or R < x , in other words, x is outside the interval [ - R,R ]. We’ll look at the behaviour of f ( x ) when i) x < - R and x → -∞ , or ii) R < x and x → ∞ . Property 1 : If deg P < deg Q then f ( x ) 0 as x → ±∞ . Proof: Write P ( x ) = n X i =0 a i x i ; a n 6 = 0 and Q ( x ) = m X j =0 b j x j ; b m 6 = 0 . We have assumed m > n . Thus dividing by x m we find for x 6 = 0 f ( x ) = (1 /x m ) P ( x ) (1 /x m ) Q ( x ) = n i =0 a i x i - m m j =0 b j x j - m . The numerator : This has the form a n x n + a n - 1 x n - 1 + ··· + a 0 x m = a n x m - n + a n - 1 x m - n +1 + ··· + a 0 x m . Since m > n each m - i > 0. Thus, as | x | → + , | x m - i | → + and a i x m - i 0. Therefore the numerator goes to 0 as x → ±∞ . The denominator : This has the form b m x m + ··· + b 0 x m = b m + b m - 1 x + ··· + b 0 x m b m as x → ±∞ . Therefore, the denominator goes to b m as x → ±∞ . The limit of f : f ( x ) = numerator denominator 0 b m = 0 as x → ±∞ because b m 6 = 0. ± Example . For f ( x ) = x 2 +10 10 x 3 + x +1 , deg(numerator) - deg(denominator) = 2 - 3 = - 1 < 0. Therefore f ( x ) 0 as x → ±∞ . Indeed,
Background image of page 2
VI.A Rational Functions as x → ±∞ VI.2 f ( x ) = x 2 +10 10 x 3 + x +1 = 1 x 3 ( x 2 +10 10 ) 1 x 3 ( x 3 + x +1) = 1 x + 10 10 x 3 1+ 1 x 2 + 1 x 3 0+0 1+0+0 = 0 as x → ±∞ . Property 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 10

goldch6 - Contents VI Asymptotic Behaviour of Rational...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online