Using Limits to Determine Big-O, Big-Omega, and Big-Theta by Stephen Clark

Are Big and Little O, , and notations properly definable using limits?
No, Big and Little O, , and notations aren't properly definable using limits.

Why is it that limits can't properly define Big and Little O, , and notations?
Limits can't properly define Big and Little O, , and notations because it's possible that even though doesn't converge to a limit.

What can you do if the ratio of and does converge or tend to infinity?
If the ratio of and does converge or tend to infinity, you can determine the asymptotic relationship between them from the value of the limit.

What are the two reasons that limits are the wrong way of thinking about asymptotic notation and expansions?
The two reasons that limits are the wrong way of thinking about asymptotic notation and expansions are:

  1. They cannot always be used.
  2. They fail to capture the true nature of asymptotic behavior even when they can be used.

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What is the definition of Big-O notation without using limits?
The definition of Big-O notation without using limits is:

What is the definition of Omega notation without using limits?
The definition of Omega notation without using limits is:

What is the definition of Theta notation without using limits?
The definition of Theta notation without using limits is:

What is the definition of Little-O notation without using limits?
The definition of Little-O notation without using limits is:

What is the definition of Little Omega notation without using limits?
The definition of Little Omega notation without using limits is:

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What does a statement of the form mean?
A statement of the form means .

Using the definition for Big-O notation above, why is ?
Using the definition for Big-O notation above, because .

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