Order notation: f(x)=O(x^n) as x-> 0

• coverband
In summary, the "big O" notation is used to describe the behavior of a function as it approaches a certain value, typically 0. It indicates that the function can be approximated by a polynomial with certain terms being negligible. This allows for a more precise understanding of the function's behavior near the given value.
coverband
Hello,

I am having difficulty getting to grips with “the order notation”. I have the following definition…

“A function f(x) is said to be of order x^n as x -> 0 if there is a non-zero constant C such that |f(x)|<C|x^n| for all x in an interval around x=0. This is written as
f(x) = O(x^n) as x->0”

some examples:

x(1+x^2)^1/2 = x + x^3/2 +… = O(x)
x/(1+x) = x(1 – x + x^2 +…) = O(x)
(x+b)^a – x^a = x^a(1 + ab/x +…) –x^a = O[x^(a-1)]

As an example, sin(x) = x + O(x^3) as x -> 0. sin(0.01) is about 0.01 - 0.01^3 * 1/6, sin(0.001) is about 0.001 - 0.001^3/6, and so forth. In this case C is something like 1/6 for an appropriate interval.

I'm not sure exactly what you want. The intuition is that the function near 0 is closely approximated by the given polynomial.

The idea behind the "big O" notation is to make rigorous the idea that, say

$$e^x = 1 + x + \mbox{ terms involving } x^2 \mbox{ or smaller}$$​

which you would write as

$$e^x = 1 + x + O(x^2) \mbox{ as } x \rightarrow 0$$​

and which means that

$$|e^x - 1 - x| < C|x^2|$$​

for some C, for all small x.

You are, in effect, treating x2 as negligible for the purpose of your calculation, but instead of just ignoring it, you keep track of what you have ignored. By the end of your calculation, you should find the ignored term goes to zero as $x \rightarrow 0$. If it doesn't, you've neglected too much, and you'll have to go back and add another term.

1. What does the notation f(x)=O(x^n) as x-> 0 mean?

The notation f(x)=O(x^n) as x-> 0 means that the function f(x) is bounded by a constant multiple of x^n as x approaches 0. In other words, the behavior of the function near 0 can be approximated by a polynomial function of degree n.

2. How is the Big O notation used in analyzing functions?

The Big O notation is commonly used in analyzing functions to describe their growth rate or order of magnitude. It provides a way to classify functions into different categories based on how quickly they grow or decrease as their input values change.

3. What is the significance of the value of n in f(x)=O(x^n) as x-> 0?

The value of n in f(x)=O(x^n) as x-> 0 represents the degree of the polynomial function that approximates the behavior of the original function near 0. A higher value of n indicates a faster growth or decrease of the function near 0.

4. Can the Big O notation be used for all types of functions?

Yes, the Big O notation can be used for all types of functions, including polynomial, exponential, logarithmic, and trigonometric functions. It is a general concept that can be applied to any function to describe its growth or decrease rate.

5. How is the Big O notation related to the concept of limits?

The Big O notation is closely related to the concept of limits, as it provides a way to describe the behavior of a function near a specific point. The limit of a function as its input approaches a certain value can be represented by the Big O notation, indicating the order of magnitude of the function's growth or decrease near that point.

Replies
3
Views
1K
Replies
3
Views
977
Replies
1
Views
853
Replies
2
Views
1K
Replies
3
Views
2K
Replies
4
Views
1K
Replies
2
Views
925
Replies
3
Views
1K
Replies
33
Views
2K
Replies
4
Views
1K