Calculus: Analytic Geometry and Calculus, with Vectors

130 Functions, limits, derivatives

that scientists of the future will adopt notation like that in (1) and that their
historians will wonder why on earth people ever concocted tales about moving
numbers and converted a few basic theorems in the theory of approximation
into a mystic "theory of limits" that kept the world agog for several centuries.
In our book, the theory of limits sometimes sounds like a theory of moving
numbers but it is in fact a part of the theory of approximation. Let us get on
with it.
5 Verify the following assertions and replace the question marks by appro-
priate answers. The basic limit theorems may be used.

(a) lim 3x = 15

(c) lim (3 - 2x) = 3

X-0

(e) lim (y + 1)(y + 2) = 20

Y-3

(g) lim x2 - x+ 1=^21

+x+1^31

(b) lira 3x =?

z-.2

(d) lim (4x - 5) =?

z-.o

(f) lim(x+2)2=?

z-.4

(h) lim

x2 - 2

=?

z- 2x2 + 2

6 Pay very close attention to the problem of evaluating

lim h - y_

h-.o h

because the process involves some troublesome points. Tell why the last part

of Theorem 3.285 cannot be used here. Look at the problem and observe that

it is difficult or impossible to guess what the answer (if any) is. Observe that

we must put the quotient in a more manageable form before we can find its limit.

The next step is to remember from experiences in algebra or to learn right now

that the numerator and denominator of the quotient should be multiplied by the

"conjugate" of the numerator. Thus

lim 2+h-1.2=lim 2+h-V-2 V2-{ h+\

h-.o h h-.o h 1V2+h+V2_

=lim 2+h-2 =lim^1

h-0 h(v'2+h+/) 2+h+ 2v'-

Tell which of the theorems of this section are used in making the last step. To

be sure that this process is thoroughly understood, make the small notational

adjustments necessary to obtain the formula

lim V;+tx1`=^1

AX--+0 Ax 21/x

Put in at least as many steps as appear in the special case.

7 Supposing that a > 0, show that

lim

x

• = 2a.
  x-.0 z-}-a2-a
  8 Show that

lim 1 + x2 - 1= 0.

z~o x