# 8 Let an 0 and [ an &lt; 80. n=1 (a) Show that liminf nan = ...

## Transcribed Text

8 Let an 0 and [ an < 80. n=1 (a) Show that liminf nan = 0. n->oo (b) Give an example showing that lim sup nan > 0 is possible. n->00 Let f : [a, b] R be continuous, and suppose that f takes on no value more than twice. Show that f takes on some value exactly once. Let T R2 R² be defined by T(u,u) = (a) Find all points where the map is locally one-to-one. Let S be the set of these points. (b) Is T one-to-one on S? (c) Determine the range of T. Let U = {(x,y) € R² x2 + y2 < 1}. Suppose that f : U R is such that both partial derivatives of f are zero at every point in U. Must f be constant? Justify your answer.

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