= Solution
As $\eta\to\infty$,
$$
f(\eta)\to1,\qquad f'(\eta)\to0.
$$
Thus the tangential velocity matches the quiescent outer fluid, but the normal velocity tends to
$$
\boxed{v\to-\alpha\delta=-\sqrt{\alpha\nu}}.
$$
The stretching sheet entrains fluid toward itself at a constant leading-order speed. Consequently a globally stationary exterior flow is impossible: the outer flow must provide this normal influx and complete the mass balance away from the local boundary-layer approximation.
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