Can You Simplify Limits Using Big O Notation?

Thread starter catapax
Start date Dec 19, 2012
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Limit Notation

In summary, big O notation is a mathematical notation used to describe the limiting behavior of a function when the input size approaches infinity. It is commonly used in computer science and mathematics to analyze the efficiency and complexity of algorithms. It is also used in limits to determine the upper bound on the growth of a function and to compare the efficiency of different algorithms. When a limit is expressed as O(n), it means that the growth rate of the function is proportional to n, indicating linear growth. Big O notation differs from big theta notation, which provides a more precise measure of a function's complexity. Big O notation is helpful in analyzing algorithms by allowing for comparison of their efficiency through their growth rates.

Dec 19, 2012

catapax

lim (e^x-sin(x)-cos(x))/(e^(x^2)-e^(x^3)), x->0
= lim (1+x+o(x)-x+o(x)+1/2(x^2)-1+o(x^2))/((x^2)+1+o(x^2)-1-(x^3)+o(x^3)), x->0
= lim 1/2(x^2)/(x^2+o(x^2)), x->0 = 1/2
is it correct?

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Dec 19, 2012

mfb

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You cannot simplify o(x)+1/2 x^2 to 1/2 x^2, as o(x) could (and does in your example!) include terms of x^2. You have to expand it more.

FAQ: Can You Simplify Limits Using Big O Notation?

What is big O notation?

Big O notation is a mathematical notation used to describe the limiting behavior of a function when the input size approaches infinity. It is commonly used in computer science and mathematics to analyze the efficiency and complexity of algorithms.

How is big O notation used in limits?

Big O notation is used in limits to describe the growth rate of a function as the input size approaches infinity. It helps to determine the upper bound on the growth of a function and can be used to compare the efficiency of different algorithms.

What does it mean when a limit is expressed as O(n)?

When a limit is expressed as O(n), it means that the growth rate of the function is proportional to n as the input size approaches infinity. This is known as linear growth, which means that the function's complexity increases at the same rate as the input size.

What is the difference between big O notation and big theta notation?

Big O notation and big theta notation are both used to describe the limiting behavior of a function, but they have different meanings. Big O notation describes the upper bound on the growth rate of a function, while big theta notation describes the tightest bound on the growth rate of a function. In other words, big theta notation provides a more precise measure of a function's complexity.

How is big O notation helpful in analyzing algorithms?

Big O notation is helpful in analyzing algorithms because it allows us to compare the efficiency of different algorithms by looking at their growth rates. A lower order of complexity (such as O(log n)) indicates a more efficient algorithm than a higher order of complexity (such as O(n^2)). This can help in making decisions about which algorithm to use in a given situation.