Book Discrete Mathematics and Its Applications 7th Edition C

Book:

Discrete Mathematics and Its Applications (7th Edition) Chapter 3.2 Question 8E

Find the least integer n such that f (x) is O(xⁿ) for each of these functions.

a)If f(x) = 2x² + x³logx then the least integer n is

b)If f(x)=3x^5+(logx)^4  then the least integer n is

c)If f(x)=(x^4+x^2+1)/(x^4+1) then the least integer n  = _____.

d) If  f(x)=(x^3+5logx)/(x^4+1) then the least integer n =

Solution

a) f(x) = 2x^2 + x^3 log x , the highest power is x^3 log x
Since we want an integer exponent, the minimal such exponent occurs when a = 1.
==> f(x) is O(x^4).

b)f(x)=3x^5+(logx)^4 , here highest power is 3x^5 , hence n =5 here

c)f(x)=(x^4+x^2+1)/(x^4+1) here approximatelythis equation is equal to 1 , since powers on both numerator and denominator are equal , hence n =1 here

d) f(x)=(x^3+5logx)/(x^4+1)    here approximatelythis equation is equal to -1 , since powers on both numerator and denominator are equal , hence n =-1 here