## limit of sequence equal to limit of function

Suppose

lim (when n -> oo) of sup {f(x) / x belongs to (0, 1/n) } = L_1, and

lim (when e -> 0+) of sup {f(x) / x belongs to (0, e) } = L_2.

(e is simply epsilon).

It seems pretty obvious, but it is truth that L_1 = L_2 ?