Following the answer to this MSE question, it is claimed that we easily show the following convergence:
$$ \text{exp}(-\sqrt{\lambda}it + \lambda (e^{it/\sqrt{\lambda}}-1)) \rightarrow_{\lambda \rightarrow \infty} \exp\left(-\frac{t^2}{2}\right)$$
Which is proved by noting that the following expression holds:
$$ \exp\left(\frac{it}{\sqrt{\lambda}} \right) -1 = \frac{it}{\sqrt{\lambda}} - \frac{t^2}{2\lambda} + o(\lambda^{-1})$$
While I understand how the former was derived, my question is how does this prove the convergence?
Since in this case,
$$\lambda (e^{it/\sqrt{\lambda}}-1) =\lambda\frac{it}{\sqrt{\lambda}} - \frac{t^2}{2} + o(\lambda^{-1/2}) $$
And I can't see why the $\sqrt{\lambda} it$ member disappears at the limit here. Does it have to do anything with the imaginary part?
What am I missing?
It follows that $$ \exp(-\sqrt{\lambda}it + \lambda (e^{it/\sqrt{\lambda}}-1))=\exp(-\sqrt{\lambda}it+\sqrt{\lambda}it-\frac{t^2}{2}+\lambda o(\lambda^{-1}))=\exp(-\frac{t^2}{2}+\lambda o(\lambda^{-1})). $$ We note that $$ \lambda o(\lambda^{-1})\stackrel{\lambda\to\infty}{\to}0 \implies \exp(-\frac{t^2}{2}+\lambda o(\lambda^{-1}))\stackrel{\lambda\to\infty}{\to}\exp(-t^2/2) $$ by continuity of the exponential function.