I was trying to solve a limit with Taylor series today, which I never had a problem with, but now I suddenly don't understand why it's ok to do it in that way.

For example, the limit

$$\begin{align}
&\lim\_{x \to 0} \frac{x - x^3/6 - \sin{x}}{x^5} \\
&= \lim\_{x \to 0} \frac{(x-x^3/6) - (x - x^3/6 + x^5/5! - \text{higher order terms} )}{x^5} \\
&= \lim\_{x \to 0} \frac{-x^5/5! + \text{higher order terms}}{x^5} \\
&= \lim\_{x \to 0} {\frac{-1}{5!} + \text{higher order terms}} = - \frac{1}{120}
\end{align}$$

But why can we say that the sum of higher order terms goes to zero? I understand that each of the terms goes to zero, but why does their *infinite* sum go to zero? How can I justify that? I think that although the summands are small, their infinite sum can hypothetically still be big?