Big-O Notation in Geometric Interpretation of Derivative

I just watched the first lecture of MIT 18.01 Single Variable Calculus, Fall 2006, and in that video, the lecturer explained about the geometric interpretation of derivative. A derivative of a function can be defined as:

The lecturer then used this to find the derivative of the function , and this is where it starts to get confusing for me. Using the definition above, the derivate of the function is equivalent to:

The part which confuses me is when the term is evaluated using the Binomial Theorem:

Why are the rest of the terms replaced by the Big-O Notation ? And why the term specifically?

Thanks! (Happy)