§ 1. Convergent sequences and series
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x + sinx ~ x when x --+ +00
since the left hand side lies between x - 1 and x + 1, so between x/2 and 2x
for x 2: 2.
(iii) The relation
f(x) rv g(x) when x --+ +00 (or when x --+ a)
will be defined in the next nO, as will be
(iv) The relation
f(x) = o(g(x»
with a lower case o. These two relations presuppose the concept of limit as
already known.
4 - The concept of limit. Continuity and differentiability
At our present level the concept of limit applies to complex-valued functions
defined on a subset X of C (and in particular of JR) when the variable x E X
increases indefinitely or else approaches a value a E C indefinitely closely. If,
for example X c JR, the relation
lim f(x) = u
x-+oo
means that, for all r > 0, one has If(x) - ul < r for x large, in other words
that, for every r > 0, there exists a number N (depending on r) such that
(4.1)
x > N ==} d[f(x), u] < r.
The limit when x tends to -00 is defined analogously: we replace the condition x > N by x < N. (One makes no assumption as to the sign of N). In
the complex case where these inequalities mean nothing one clearly needs to
write that
Ixl > N ==} If(x) - ul < r.
Similarly, the relation
lim f(x) = u
x-+a
means that, for all r > 0, one has
(4.2)
d[f(x), u] < r for all x E X sufficiently close to a
i.e. that there exists a number r' > ° (depending on r) such that, for x EX,
the relation
(4.3)
Ix - al < r' implies If(x) - ul < r.
Note in passing that we do not assume that a E X: one can have X =]0,1[
and a = 0. For example, in X = C, liz tends to ° when z --+ 00 since the