A student claims that the limit _ x ( x^ 2 +7x -x ) must diverge because each…
MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
A student claims that the limit
x→∞lim(x2+7x−x)
must diverge because each individual term grows without bound as x becomes large.
(1) Explain, without explicit computation, why the student’s reasoning is incomplete. In your explanation, refer to the relative growth rates of the two terms inside the limit.
(2) Describe a systematic algebraic procedure that can reveal whether the limit is finite, infinite, or nonexistent when faced with a difference of two unbounded expressions of the same order.
(3) Using the reasoning you outlined in part (b), predict qualitatively (increasing, decreasing, or leveling off) how the graph of y=x2+7x−x behaves for large positive x. Justify your description based on end-behavior analysis.