Sequence converging to different limits

In a metric space, is it possible to find a sequence which converges to two different limits with respect to two different metrics?

Obviously the metrics can't be equivalent.

With two different metrics? Yes, obviously. (But with two different metrics it is not the same metric space, by definition -- the concept of a metric space includes which metric we're using).

For example take $\mathbb R$ with respectively the standard metric, and the metric $$ d_2(a,b)=|f(a)-f(b)| \quad\text{where }f(x)=\begin{cases} \pi & \text{if }x=0 \\ 0 & \text{if }x=\pi \\ x & \text{otherwise} \end{cases} $$

Then $a_n=\frac1n$ converges to $0$ in the usual metric but to $\pi$ in the $d_2$ metric.