Calculus: Analytic Geometry and Calculus, with Vectors

3.3 Unilateral limits and asymptotes 137

cosx lim L1 x2 x4-2i+¢i-6i+x3 +(_1)n(2n)!1x2n

(v) n-. w .11

r x3 xs x7

n

x2nt1

(a) sinx= lim +(-1)n(2n+1)!Jl

-, w

\

(x) sin ax = lim C7rx (1- is)(1- 22/(1 - 32).. (1- ns/Jl 1

(y) x! = lim L \ n!e / \ (x - 1, - _, - 3,

nw (x + 1)(x + 2)(x + 3)... (x + n)

(a) lim x!

x-+-1+

2 Does the statement

approx 1 = 0

e,n>N^71

.)

abbreviate the statement that to each positive number e there corresponds an
integer N such that Il/ni < e whenever n > N? fins.: It can, but it does only
if we agree that it does. Remark: Whether the above abbreviation is better than
the abbreviation

lim l =0

n-.-

is purely a matter of opinion. If a person has the habit of using one notation,
the other must seem to be quite absurd, awkward, and unteachable.
3 Draw a graph of the equation y = x2. Then, supposing that 11 is a given
number, show how the figure can be used to support the assertion that

lim x2 = oo.

Z-

4 4 One of the assertions

lim 1 = 0, (?), lim l ao (?)

X-0 x x-0

is true and the other is false. Give a full discussion of this matter. Remark:
Here and elsewhere, displayed assertions followed by question marks may be
false assertions.
5 With the aid of the idea that the numerator and denominator of the first
quotient can be divided by x, show that

(a) wx - 1= 1

x + I

lim x2+2x+3 =^1

-.,2x2-2x+3 2

(d)

(f) lim

x2s-

+^11 -

2 (c) =limes 3x3+ 1 =^3

(e) limx2 + 2x + 3-^0

x-.wx3-2x+3

(b) lim

x-.w

(g) lim x =

x-.0 1/1 + x

6 Show that both coordinate axes are asymptotes of the graph of the equa-

tion y = 1/x.

7 There will come a day when we must learn that the graph of the equation

x2 y2_

a2-bs -1.