In our lecture notes, we demonstrated using limit to analyze the order of growth for functions in algorithm analysis (on slides 21 and 22). Please explore a bit more into algorithm analysis, and list 3 to 5 concepts/methods where the notion of limit is used. In your opinion, can any of these concepts/methods be accomplished in ways other than limits? Why or why not?

21 Analysis of two functions using limits Check Point! Algorithm analysis Checkpoint 1 Prove that if f (x) = 4x3 + 3x2 and g(x) = x4, then f(x) E o(g(x)) We can set up a limit quotient between to f (n) and g(:) as folows to determine the order of growth for these two functions: Solution: lim (2) = lim 4x3+3x2 thea /(m) en((w)) n-cog(x) n-+00 x4 - = lim (+ 3x2 n-co *4) = lim #+=0 The Nals fisell notation means a() is an asymptotic lower bour The fiske eviell notation means that /(it) and g be) are of the same order of growth. according to 22 lim f (n) 0 then f(n) E O(g(n)) Check Point! ( 2 ) 6 00-u c> 0 then f(n) E O(g(n) Prove that if f(x) - 4x' + 3x* and g(x) = x*, then /(x] c o(9(x)) 00 then f(n) en(g(n)) Solution: lim - lin - In (+ !)- Im (+ ;) -0 according to so f (x) E o(g(x)) so f(x] c D(g(x]) 22 23 Continuity 22 finition of Continuity Let e be a real number in the interval for. bk and let f he a function whose domain are the. seetales the interval (e, b). The fuaction / h continwon at f when these conditions 1. Ach is defined If f is continuous at every point in the interval Co. My. then f is centi open interval la, bk. ous on the intiruity of Polynomial and Rational Functions VO: Checkpoint. 1. A polynomial function is continuous at every real number. 2. A rational function is continuous at every member in its dorruin. Find the solution to the equation. 24 Continuity on a Closed IntervalBA]: LE(E) Solution: 20 21 Analysis of two functions using limits Analysis of two functions using limits Algorithm analysis Algorithm analysis We can set up a limit quotient between to / (1) and g[ determine the order of growth for these two functions: 20 then fals Algal then /(m) e f(s(n)) therehay field under bound for f () The Nel chisel notation means g() is an asymptotic lowe The fiske exelelf notation means that /(it) and g(#) are of the same order of growth. 22 Check Point! We can set up a limit quotient between to f (n) and g(n) as follows to determine the order of growth for these two functions: Prove that if f'(x) - 4x' + 3x* and g(x) - x*, then /(x] c o(9(x)) Solution: according to 0 then f(n) E O(g(n)) lim f(n) n-+0 g(n) c> 0 then f(n) E O(g(n)) then / (m) e f(s (w)) then f(n) E (g(n)) so /(x] c D(s(x)) The f(n) E O(g(n)) notation means g(n) is an asymptotic upper bound for f (n). 22 The f(n) E 2(g(n)) notation means g(n) is an asymptotic lower bound for f (n) The f(n) e e(g(n)) notation means that f (n) and g(n) are of the same order of growth. 23 21 Continuity Definition of Continuity Let e be a real number in the interval fa, b), and let f be a function whowe domain are true. catalina the interval (o, b). The function / in continuon at & when these conditions 1. Mich is defined If f is continuous at every point in the interval do. M. then f is continuous on the open interval ta, b1. and Ratio VO: Analysis of two functions using limits