Calculus: Analytic Geometry and Calculus, with Vectors

136 Functions, limits, derivatives

In this case, the line having the equation y = 4x + B is called an

asymptote of the graph. This asymptote is horizontal if .1 = 0. We
want to be able to apply similar jargon to graphs of equations, such as

(3.38)

x2 y2-1

a2 b2

which are not graphs of functions. When we start with an equation of the
form (3.38) and transpose all of the terms to the left side, we obtain an
equation of the form

(3.381) F(x,y) = 0.

If f is a function such that (3.381) is true when y = f(x), then each

asymptote of the graph of y = f(x) is also an asymptote of the graph of

(3.381). Problem 7 at the end of this section involves the famous

equation (3.38).

Problems 3.39

1 Using epsilons appropriately, give a full statement of the meaning of

each of the following truthful assertions. In case an assertion is so subtle that

we are not yet prepared to prove it and appreciate its consequences, we need not

be disturbed. Scientists can, for example, understand the assertion "there is

helium in the sun" before they are able to prove the fact and understand the role

of helium in the production of energy radiated by the sun.

(a) lim^1 = 0

x-+mx

(c) lim^1 = 00

x-+0+ X

(e) lim N / ' - x 2 = 0

(g)

(i)

(k)

lim tan x = m

x r/2-

lim log x = 00

a-+m

lim e- = 0

(u) ez = lim CI

3!

+

(1)

lim

lim

xlog(I+1)

X

el = co

(t) Urn 1x}" = 0, (}xJ < 1)

n-..o

nx