List of some basic formulas and concepts related to infinitesimal functions:
- Definition of an infinitesimal function: A function ( f(x) ) is called infinitesimal at ( x \to a ) if ( \lim_{x \to a} f(x) = 0 ).
- Comparison of infinitesimals: If ( f(x) ) and ( g(x) ) are infinitesimal functions as ( x \to a ), then:
- ( f(x) = o(g(x)) ) means that ( \frac{f(x)}{g(x)} \to 0 ) as ( x \to a ).
- ( f(x) = O(g(x)) ) means that there exists a constant ( C ) such that ( |f(x)| \leq C |g(x)| ) as ( x \to a ).
- Examples of infinitesimal functions:
- ( f(x) = x – a ) при ( x \to a )
- ( f(x) = \sin(x) ) при ( x \to 0 )
- ( f(x) = e^x – 1 ) при ( x \to 0 )
- Properties of infinitesimal functions:
- The sum of two infinitely small functions is also infinitely small: if ( f(x) ) and ( g(x) ) are infinitely small, then ( f(x) + g(x) ) is infinitely small.
- The product of an infinitesimal function and a bounded function is also infinitesimal: if ( f(x) ) is infinitesimal and ( g(x) ) is bounded, then ( f(x)g(x) ) is infinitesimal.
- Infinitesimal Theorem: If ( f(x) ) is an infinitesimal function, then ( f(x) ) can be represented as a Taylor series expansion.