$o$ notation for different variables at the same time?

Question

In a proof for the Central Limit Theorem using characteristic functions, one can expand $\varphi$ into a Taylor series and get $$\varphi_n(t):=\varphi_{\frac{S_n}{\sqrt{n}\sigma}}(t) = \left(1-\frac{t^2\sigma^2}{2n\sigma^2}+o\left(\frac{t^2}{n^2}\right)\right) \quad \text{as }t\to 0 \quad (1)$$ In the following step it is concluded that $$\lim_{n\to \infty}\varphi_n(t) = e^{-t^2/2}.$$ Now how do you formally justify that
$1)$ "$o(t^2/n^2) \to 0$ as $n\to \infty$"
2) I know that $$\left(1+\frac{x}{n}\right)^n \to e^x.$$ I don't see how this immediately justifies $(1)$, since the expression $o(t^2/n^2)$ still depends on t and you can't just ignore it, can you? Maybe someone can clarify!

Answer 1

Not quite. Using Taylor expansion, one gets \begin{align} \phi_n = \left( 1 - \frac{t^2}{2n} + o\left(\frac{t^2}{n}\right) \right)^n, \quad \frac{t^2}{n} \to 0. \end{align}