# Infinitesimals' rates of approaching 0

• I
• feynman1
In summary, determining the speed or rate at which two infinitesimals approach zero can be done by comparing their ratios or magnitudes. The definition of speed may vary depending on the context and the specific question being asked.

#### feynman1

When comparing 2 infinitesimals, does the higher order one approach 0 faster or slower?

Why don't you take the ratio of the two to see it ?

anuttarasammyak said:
Why don't you take the ratio of the two to see it ?
I know how to do the maths, here I'm asking just about the statement.

##x^3## goes to zero faster than ##x^2##, ##|x^3|<|x^2|##, when x of -1< x < 1 approaches to zero.

anuttarasammyak said:
##x^3## goes to zero faster than ##x^2##, ##|x^3|<|x^2|##, when x of -1< x < 1 approaches to zero.
how did you define speed/fast

I am sorry to say I do not have confidence on my rigorous use of mathematics and English how to express ##|x^3|<|x^2|## for ##|x|<1## which approaches zero or is numerical sequence ##x_n## such that
$$\forall \epsilon>0\ \exists N \ \ N<n\ \ |x_n|<\epsilon$$

Last edited:
feynman1 said:
how did you define speed/fast

Usually speed is defined by looking at a ratio or just comparing magnitudes If ##f(0)=g(0)=0## and f and g are continuous at 0, we could say ##g(x)## goes to 0 faster than ##f(x)## if ##|g(\epsilon)| < |f(\epsilon)| ## for all sufficiently small ##\epsilon##. Sometimes you want to break ties a bit blunter - e.g. you might want to say that ##\sin(x)## and ##x## go to 0 at the same speed, so you would want ##|g(\epsilon)/f(\epsilon)| < 1## to say ##g## goes to zero faster. Sometimes you really want things like ##x## and ##2x## to count as going to 0 at the same speed, in which case you might require the ratio to go to 0 as ##\epsilon## goes to 0. It really depends on the context and why you are trying to pick one thing going to zero faster than the other.

I would say the most common context is you have several terms you're adding together and you want to ignore one entirely, in which case you would probably say ##g## goes to 0 faster than ##f## if ##\lim_{x\to 0} g(x)/f(x)=0##

feynman1

## 1. What are infinitesimals?

Infinitesimals are mathematical objects that are infinitely small, but not equal to zero. They are used in calculus to represent quantities that approach zero, but never actually reach it.

## 2. How are infinitesimals used in calculus?

Infinitesimals are used in calculus to represent rates of change, such as the slope of a curve or the velocity of an object. They allow us to analyze and solve problems involving continuously changing quantities.

## 3. Can infinitesimals be negative?

Yes, infinitesimals can be both positive and negative. They are simply numbers that are infinitely close to zero, so they can take on any value on the number line.

## 4. Are infinitesimals the same as limits?

No, infinitesimals and limits are not the same. Infinitesimals are used in non-standard analysis, while limits are used in standard analysis. Infinitesimals are also defined as actual numbers, while limits are a concept used to approach a value.

## 5. Why are infinitesimals important in calculus?

Infinitesimals are important in calculus because they allow us to analyze and understand continuously changing quantities. They also help us to solve complex problems in physics, engineering, and other fields that involve rates of change.