(e) Give an example of two nonlinear functions that do not dominate each other.

Question

(e) Give an example of two nonlinear functions that do not dominate each other.

5. Computer scientists and chaos theorists often compare functions and their growth rates. Some functions grow at a seemingly similar rate while others grow at very different rates. When one function grows way faster than another, we say it dominates the slower function. Mathematically, we say that g dominates flar g(x) ,iflin!(r) = linxg(r)-x and either lin“ g r)=0orlin -oo. (a) Which function dominates the other? Justify your conclusion: In r or vr. lim g(x)=oo and either lim =0 or lim (b) Which function dominates the other? Justify your conclusion: In or vT, where n e Z,n >0 (c) Explain why ez will dominate any polynomial function. (d) Explain why x" will dominate In r for all large values of n

Explanation / Answer

a)Exponential function growth is faster than polynomial, However logrithm growth is slower than rdical function.
So sqrt(x) is dominates to ln(x).

(b) It depends on the value of n , which is dominating ln(x) or (x)^(1/n).
e=2.71, so if n> 2.71, then (x)^(1/n) will dominate ln(x) . And for n<=2.71 ln(x) is dominating .

(c) After some instance e^x value will dominate polynomial. Because it's base is 2.71 and it always increase with 2.71 times.

(d)

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