# Which function goes faster?

minase
If you are given two functions and asked which function goes faster. You divide on function by the other and you use lopitals rule. If your ratio gives you a constant it is said the two fuction grow at the same rate. I can't understand why 5x and 2x would be growing at the same rate. 5=2? Isn't rate 5 for 5x and 2 for 2x. how is it 5x and 2x growing at the same rate?

sutupidmath
remember that we are talking for infinitely large or small terms, so the fact that 5>2 does not make any change, since lim(5x/2x)=5/2, when x->infinity, so actually here f(x)=5x and f(x)=2x grow at the same rate. Because 5* infinity is also infinity,and 2* infinity is also infinity, so 5, and 2 and any other constand do not really make any change. They would not grow at the same rate if we would have let's say f(x)=5x^3 and f(x)=2x, in this case f(x)=5x^3 , would grow at a faster rate, since the limit of their ratio when x-> infinity,would give you either 0 or infinity, depends whic function you take as a denominator,and which as a numerator.

anyone correct me if i am wrong!!

Homework Helper
"Which function goes faster" makes no sense. I think you are asking "which function goes to infinity faster" which can be determined by
[tex]\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}[/itex]
If that limit is a finite, non-zero, number, then f= O(g) (and g= O(f)). If it is 0, then f= o(g). If it is infinite, then g= o(g).

tehno
If you are given two functions and asked which function goes faster. You divide on function by the other and you use lopitals rule. If your ratio gives you a constant it is said the two fuction grow at the same rate. I can't understand why 5x and 2x would be growing at the same rate. 5=2?
I agreee that's kinda stupid terminology .Get used to it.
It makes more sense to me to say f1(x)=5x grows "faster" than f2(x)=2x for $x\in\mathbb{R^+}$.

sutupidmath
If it is 0, then f= o(g). If it is infinite, then g= o(g).
at the last one i think halls meant
g= o(f), since at this case g(x) is said to be of a higher order than f, considering that both of them are infinitely small.

sutupidmath
HallsofIvy;1285869[tex said:
\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}[/itex]
If that limit is a finite, non-zero, number, then f= O(g) (and g= O(f)). If it is 0, then f= o(g). If it is infinite, then g= o(g).

yeah and if f, and g are infinitely large when x->infinity, than it is vice verse

Homework Helper
yeah and if f, and g are infinitely large when x->infinity, than it is vice verse

Actually, I was assuming that f and g are unbounded so I'm not sure what you mean by "vice versa".

Staff Emeritus