Given two functions which \"go to zero\", which of the two approaches zero \"fas

Question

Given two functions which "go to zero", which of the two approaches zero "faster"? For example, by looking at the graphs below (concave up power functions with p > 1, k = 1) we could say something like "As x right arrow 0, the higher the power, the faster it approaches zero." A specific example: compare f(x) = 0.25x: and g(x) = 500x^2 for x closer and closer to 0: F(0.01) = and G(0.01) = which will give us the wrong impression since lim x right arrow 0 g(x)/f(x) = 0 lim x right arrow 0 500x^2/0.25x = lim x right arrow 0 500/0.25 .x^2/x = lim x right arrow 0 2000 x = the function goes to zero faster than the function Even dividing by something small (something approaching zero), it's still able to bring the ratio to zero. Below are the graphs of the functions y = x,x^l.5,x^2,x^2.5,x^3 and x^5.5 in two different scales.

Explanation / Answer

f(0.01) = 0.25*0.01 = 0.0025

g(0.01) = 500*0.012 = 0.05

Lt x tends to 0 f(x)/g(x) = 0

The function f(x) goes to zero faster than the function g(x)

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