Consider two polynomials and of degrees and respectively

MATHEMATICS · Subjective · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Consider two polynomials

P(x)

and

Q(x)

of degrees

m

and

n

respectively, where

m,n\ge 1

. (i) State the rule that determines whether

\displaystyle\lim_{x\to\infty}\frac{P(x)}{Q(x)}

0

, a finite non-zero constant, or does not exist as a finite real number. (ii) Explain, without performing any numerical substitution, why the limit of

\dfrac{7x^{5}+3x^{2}-1}{4x^{6}-2x+9}

0

. (iii) Let

R(x)=ax^{3}+bx^{2}+1

and

S(x)=cx^{3}+5x^{2}-4

. Describe clearly the conditions that

a

and

c

must satisfy so that

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

exists and equals a finite non-zero real number.

Related theory

Limits
Limits of algebraic functions
Limits and Derivatives
Limits and Continuity

Browse the problem library