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

Section 2.3 Calculating Limits Using the Limit Laws 97
More generally, we have the following law, which is proved in Section 2.5 as a consequence
of Law 10.
Root Law
Newton and Limits
Isaac Newton was born on Christmas
Day in 1642, the year of Galileo’s death.
When he entered Cambridge University
in 1661 Newton didn’t know much
mathematics, but he learned quickly
by reading Euclid and Descartes and by
attending the lectures of Isaac Barrow.
Cam bridge was closed because of the
plague in 1665 and 1666, and Newton
returned home to reflect on what he
had learned. Those two years were
amazingly productive for at that time
he made four of his major discoveries:
(1) his repre senta tion of functions as
sums of infinite series, including the
binomial theorem; (2) his work on differential
and integral calculus; (3) his laws
of motion and law of universal gravitation;
and (4) his prism experi ments on
the nature of light and color. Because of
a fear of controversy and criticism, he
was reluctant to publish his discoveries
and it wasn’t until 1687, at the urging
of the astronomer Halley, that Newton
published Principia Mathematica. In
this work, the great est scientific treatise
ever written, Newton set forth his version
of calculus and used it to investigate
mechanics, fluid dynamics, and
wave motion, and to explain the motion
of planets and comets.
The beginnings of calculus are
found in the calculations of areas and
volumes by ancient Greek scholars such
as Eudoxus and Archimedes. Although
aspects of the idea of a limit are implicit
in their “method of exhaustion,” Eudoxus
and Archimedes never explicitly formulated
the concept of a limit. Like wise,
mathematicians such as Cavalieri, Fer -
mat, and Barrow, the immediate precursors
of Newton in the development of
calculus, did not actually use limits. It
was Isaac Newton who was the first to
talk explicitly about limits. He explained
that the main idea behind limits is that
quantities “approach nearer than by
any given difference.” Newton stated
that the limit was the basic concept in
calculus, but it was left to later mathematicians
like Cauchy to clarify his ideas
about limits.
11. lim
x la sn f sxd −
s n
lim f sxd
x la
where n is a positive integer
f If n is even, we assume that lim f sxd . 0. g
x la
ExamplE 2 Evaluate the following limits and justify each step.
(a) lim
x l5
s2x 2 2 3x 1 4d
SOLUTION
(a)
(b) lim
x l22
x 3 1 2x 2 2 1
5 2 3x
lim s2x 2 2 3x 1 4d − lim s2x 2 d 2 lim s3xd 1 lim 4 (by Laws 2 and 1)
x l5 x l5 x l5 x l5
− 2 lim x 2 2 3 lim x 1 lim 4 (by 3)
x l5 x l5 x l5
− 2s5 2 d 2 3s5d 1 4 (by 9, 8, and 7)
− 39
(b) We start by using Law 5, but its use is fully justified only at the final stage when we
see that the limits of the numerator and denominator exist and the limit of the denominator
is not 0.
x 3 1 2x 2 2 1
lim
−
x l22 5 2 3x
−
lim sx 3 1 2x 2 2 1d
x l22
lim s5 2 3xd (by Law 5)
x l22
lim 1 2 lim 2 lim x l22 x l22 x l22
lim x
x l22 x l22
(by 1, 2, and 3)
− s22d3 1 2s22d 2 2 1
5 2 3s22d
− 2 1
11
(by 9, 8, and 7)
NOTE If we let f sxd − 2x 2 2 3x 1 4, then f s5d − 39. In other words, we would
have gotten the correct answer in Example 2(a) by substituting 5 for x. Similarly, direct
substitution provides the correct answer in part (b). The functions in Example 2 are
a polynomial and a rational function, respectively, and similar use of the Limit Laws
proves that direct substitution always works for such functions (see Exercises 57 and 58).
We state this fact as follows.
Direct Substitution Property If f is a polynomial or a rational function and a is
in the domain of f , then
lim f sxd − f sad
x la
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