Let be any polynomial of degree with real coefficients and let be a positive real number.

MATHEMATICS · Multi-part · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

Let

P(x)

be any polynomial of degree

n\ge 1

with real coefficients and let

a

be a positive real number. (i) Rewrite

P(x)

in the form

x^{n}Q(x)

where

Q(x)

approaches a non-zero constant as

x\to\infty

. (ii) Using the expression obtained in part (i), prove that

\displaystyle \lim_{x\to\infty} P(x)e^{-ax}=0

. Your proof must be self-contained and should not appeal to any result beyond the definition of limit and basic limit laws. (iii) Hence evaluate

\displaystyle \lim_{x\to\infty} \dfrac{P(x)}{e^{ax}}

. State clearly any conclusion you draw about the comparative growth of polynomial and exponential functions.