Calculus: Analytic Geometry and Calculus, with Vectors

3.2 Limits 129

Problems 3.29

1 It is not enough to be able to read the four assertions which involve f(x)
when x is near a but x 0 a. We must be able to write them. Try to write
them with the text out of sight and, if unsuccessful, read the text some more and
try again.
2 In terms of epsilons and deltas, write a complete statement giving the
exact meaning of each of the following true statements:

(a) x3 is near 27 whenever x is near 3 but x 0 3.

(b) sin x is near 0 whenever x is near 0 and x 0.

(c) sin x is near 1 whenever x is near 0 and x 9d 0.

x

(d) 1 - zos x isnear 0 whenever x is near 0 and x 5A 0.

(e) x is near 1 whenever x is near 1 and x 3 1.

(f) lim

sin x= 1 (g) liml - cos x= 0

x- O X x-.0 X

(h) lim

1 - cosx= (^1) (i) lim ex = e2
x-i0 x2^2
(j) lim (1 + x) 112 = e (k) lim
ex- 1 = 1
X-0 x-0 X
(1) lim
sin (x + Ox) - sin x=cos z
Ax-+o Ax
(m) lim cos (x + Ox) - cos x sin x
9x-.0 Ax
(n) lim nx"+' - (n + 1)x" + 1
=n(n + 1)
x-+1 (x - 1) 2^2
Answer to last part: To each E > 0 corresponds a 6 > 0 such that

nx"+1 - (n+1)x"+1 -n(n+1) <

I (x - 1)2^2

whenever 0 < Ix - 11 < 6.

3 The first formula of Theorem 3.285 assures us that if f(x) is near 3and

g(x) is near 5 when x is near a but x 0 a, then f(x) + g(x) is near 8 whenever x

is near a but x 0 a. Give similar applications of the other two formulas in the

theorem.

4 Tell whether you would like to learn and use new notation by which one

or the other of the "formulas"

(1) approx f(x) = L, approx f(x) = L

e,0<jx-aj<5 x=a

is used to abbreviate the epsilon-delta assertion: to each positive number e

there corresponds a positive number 6 such that f(x) approximates L so closely

that lf(x) - Ll < e whenever 0 < Ix - al < 3. If you have no opinion, think

about the matter and get one. Remark: A person who thinks that this is a

silly question may be thoroughly mistaken. It is not unreasonable to suppose