James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 3.3 Derivatives of Trigonometric Functions 195
ExamplE 4 Find the 27th derivative of cos x.
SOLUtion The first few derivatives of f sxd − cos x are as follows:
PS Look for a pattern.
f 9sxd − 2sin x
f 99sxd − 2cos x
f999sxd − sin x
f s4d sxd − cos x
f s5d sxd − 2sin x
We see that the successive derivatives occur in a cycle of length 4 and, in particular,
f snd sxd − cos x whenever n is a multiple of 4. Therefore
f s24d sxd − cos x
and, differentiating three more times, we have
f s27d sxd − sin x
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Our main use for the limit in Equation 2 has been to prove the differentiation formula
for the sine function. But this limit is also useful in finding certain other trigonometric
limits, as the following two examples show.
Note that sin 7x ± 7 sin x.
sin 7x
ExamplE 5 Find lim
x l 0 4x .
SOLUtion In order to apply Equation 2, we first rewrite the function by multiplying
and dividing by 7:
sin 7x
4x
− 7 4S
sin 7x
7x
If we let − 7x, then l 0 as x l 0, so by Equation 2 we have
sin 7x
lim
x l 0 4x
− 7 4 lim
x l 0S
− 7 4 lim
l 0
sin 7x
7x
sin
D
D
− 7 4 ? 1 − 7 4
ExamplE 6 Calculate lim x cot x.
x l 0
SOLUtion Here we divide numerator and denominator by x:
x cos x
lim x cot x − lim
x l 0 x l 0 sin x
cos x
− lim
x l 0 sin x
x
− cos 0
1
− 1
−
lim cos x
x l 0
lim
x l 0
sin x
x
(by the continuity of cosine and Equation 2)
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