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AP Calculus Question of the Month: November
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by Ben Klein
Davidson College
Davidson, North Carolina
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Theorems About Limits with Applications
We begin this Question of the Month by asking to prove some general theorems about limits. Then we will ask you to apply these results to find some specific limits involving trigonometric functions.
In parts (A), (B), and (C), f denotes a function whose domain includes an interval that contains 0. Part (A) is very easy; we have included it as a contrast to part (B).
(A) Find a function f and a nonzero real number r such that
, but 
.
(B) Suppose that 
for some positive integer p. Show that if r is a nonzero real number, then 
.
(C) Suppose that 
for some positive integer p. Show that if q > 1 is an integer, then 
.
Before asking you to apply (B) and (C), we note that it is possible to relax the assumption that p and q are integers, if we take limits from the right at 0, instead of two-sided limits. If we do consider more general p and q, we must be careful about the sign of r.
We will now ask you to apply (B) and (C) to establish some results about limits involving trigonometric functions. None of these results are new; almost all of them should be familiar. However, the derivations here are different from the ones you will see in textbooks. Many of the results can be established using l'Hôpital's rule, but our intent here is to obtain the results using much more elementary methods.
We assume that the sine and cosine functions are continuous so that:
We also assume the following result. A proof based on area can be found in almost any calculus textbook, typically in the chapter in which the derivatives of the trigonometric functions are derived.
(D) Apply the results in (B) and (C) above to show that if r is a nonzero real number and if q > 1 is an integer, then 
and 
.
(E) Use the fact that 
and the limit in (*) above to find the value of 
.
(F) Use the fact that 
and the result in part (D) to verify your answer to part (E).
(G) Use your answer to part (E) to find the values of the following limits, assuming that r is nonzero and that q > 1 is an integer:
.
(H) Assume that 
exists and use the identity
to evaluate the given limit.
Click here to view the answers and commentary!
Ben Klein is currently the Beverly F. Dolan Professor of Mathematics at Davidson College in Davidson, North Carolina, where he has taught since 1971. Ben's relationship with AP Calculus began in 1990 when he served as a Reader at Clemson University. He has attended every Reading since then and has served as a Table Leader in recent years. In 2003 he completed a four-year term on the AP Calculus Development Committee.
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