Why Are 5x and 2x Considered to Grow at the Same Rate?

[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!

 

[HallsofIvy]

"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]

\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

 

[HallsofIvy]

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".

 

[Hurkyl]

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?

You should consider "the same rate" a technical term with a technical definition that may or may not relate to a common English interpretation of those same words.

We use this definition because it's useful. Constant factors are often irrelevant. In analysis, there is no difference between "x goes to zero" and "4x goes to zero". In computer science, we choose up front to ignore the constant factors, so that we can prove things in generality without having to worry about the architectural differences between different computers, or the optimization skill of the programmer / compiler, et cetera.