# goldch6 - Contents VI Asymptotic Behaviour of Rational...

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

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

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

View Full Document
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,
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 : If P ( x ) = n i =0 a i x i , a n 6 = 0 and Q ( x ) = n j =0 b j x j , b n 6 = 0, have same degree then f ( x ) a n b n 6 = 0 as x → ±∞ . Proof: Using long division, we divide Q ( x ) into P ( x ), to get P ( x ) = a n b n Q ( x ) + R ( x ) , with deg R < n , or R = 0.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern