Consider the general rational function R(x)= a_nx^ n +a_ n-1 x^ n-1 + +a_1x+a_0 b_mx^…

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

Consider the general rational function

R(x)=\frac{a_nx^{n}+a_{n-1}x^{n-1}+\dots+a_1x+a_0}{b_mx^{m}+b_{m-1}x^{m-1}+\dots+b_1x+b_0},

where

a_n\neq 0

b_m\neq 0

, and

n,m

are non-negative integers. Analyze the long–run behavior of

R(x)

x\to\infty

in the three distinct degree relationships between numerator and denominator. (a) When

n<m

, describe the value approached by

R(x)

x\to\infty

and give a clear justification based on the structure of the function. (b) When

n=m

, determine

\displaystyle\lim_{x\to\infty}R(x)

in terms of the leading coefficients

a_n

and

b_m

. Explain each algebraic step in your reasoning. (c) When

n>m

, explain why

\displaystyle\lim_{x\to\infty}R(x)

diverges. Discuss how the signs of

a_n

and

b_m

influence whether the limit is

+\infty

-\infty

. Provide a logical argument rather than specific numerical computation.