No, such a sequence is not posible.
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If I understand your question correctly, such a sequence is an = x cos(πx). It has neither an upper nor lower bound. It's divergent, but its limit is neither infinity nor negative infinity.
Students surely can recognize the number that is the limit of this sequence.
The limit of cos2(x)/x as x approaches 0 does not exist. As x approaches 0 from the left, the limit is negative infinity. As x approaches 0 from the right, the limit is positive infinity. These two values would have to be equal for a limit to exist.
The limit of the ratio is the Golden ratio, or [1 + sqrt(5)]/2
As x goes to infinity, the limit does not exist.