- Thread starter
- #1

#### ATroelstein

##### New member

- Jun 30, 2012

- 15

Under Calculation of Big - Oh, some theorems are provided that classify the growth rates of functions in relation to one depending on what the limit is as the input approaches infinity. One proof is provided when the limit is a constant using the definition of a limit, but I'm trying to also prove the others ones, such as when the limit is 0. I came up with the following proof, skipping restating some of the definition that has already been provided in the example proof.

$$

\frac{f(x)}{g(x)} - 0 < \epsilon

$$

$$

\epsilon = 1

$$

$$

\frac{f(x)}{g(x)} - 0 < 1

$$

$$

\frac{f(x)}{g(x)} < 1

$$

$$

f(x) < 1 * g(x)

$$

Which shows 1 is the constant and N0 is the N that I need to then prove this is correct using the definition of Big Oh. My question is, does my proof flow correctly? Also, are you technically able to pick any positive value for epsilon, or is there a technique involved with selecting this value. Lastly, what would the proof look like if the limit was infinity? I'm confused when I set up the initial part of that proof as the f(x)/g(x) would be minus infinity. Thanks in advance for any guidance.