Big O: Clarifying Prime Obsession Concepts

[2710]

hi,

I am reading Prime Obsession and john talks about Big o. He goes if Function A is trapped within + and - of function B then F(A) is Big O of F(B)

So, y=Sin x is Big O of Y = 1 right?

He also goes that if F(A) is Big O of F(B) then so is any multiple of F(A).

so y=2sinx is also big O of Y=1? Even though is breaks the limit infinite amount of times?

Any clarification appreciated. Thanks!

 

[CompuChip]

Yes, so the definition is not: "A is trapped between + and - the value of B", but "A is trapped between + and - a multiple of the value of B".

Technically speaking, we say that
$$f(x) = \mathcal O(g(x)) \qquad (x \to \infty)$$
if there exists some number M > 0 and some value x₀ such that
$$x > x_0 \implies |f(x)| < M |g(x)|$$

If you have f(x) = c sin(x), then you can easily show that f(x) = O(c), where c is the constant function which takes value c everywhere.
Note that even g(x) = c sin(x) / xⁿ is O(c) (for all n > 1), because for x > 1 we have |g(x)| < |f(x)| < c.

Analogously to the definition of limit as x goes to infinity vs. limit as x goes to some finite number a, we can also define
$$f(x) = \mathcal O(g(x)) \qquad (x \to a)$$
if there exists some number M > 0 and some value $\delta > 0$ such that
$$|x - a| < \delta \implies |f(x)| < M |g(x)|$$

 

[2710]

oh yeh, forgot about that. Thanks! :D