In the GeoGebra application, $e \approx (1+ \frac 1 {2^k})^{2^k}
$. Notice how the arrows on the mapping diagrams are paired with
the points on the graph.
You can use the slider to move the point for $k$ on the mapping
diagram to see how the function value for the estimate for $e$
changes both on the diagram and on the graph.