from x= 1,000,000 onwards. In fact, for any small value
εit is possible to locate a value x= Nso that all outputs
of the function from that position onward are within a
distance εfrom zero. (For example, N= will do.)
If a function f(x) has a limit value Las xbecomes
large, we write: limx→∞ f(x) = L. One can similarly
define the notion of a limit as xbecomes large and neg-
ative: limx→–∞f(x) = L.
One can also consider the possibility of the outputs
of a function f(x) approaching a value Las xapproaches
a finite value a. Loosely speaking, we say the “limit of
f(x) as xtends to ais L” if, as xgets closer and closer to
a, the outputs f(x) get closer and closer to L. If this is
indeed the case, we write: limx→af(x) = L.
To make this notion precise, we need to assume
one is given a desired degree of precision ε, and show
that it is indeed possible to specify a “degree of close-
ness to a” that ensures all the outputs f(x) are within a
distance εof L. This leads to Weierstrass’s famous
epsilon-delta definition of a limit:
A function f(x) has “limit Las xtends to a” if
one can demonstrate that for any positive
number ε(no matter how small), the outputs
of the function can be made this close to Lby
restricting xto values very close, but not equal,
to a. That is, one can produce a number δso
that if x, different from a, lies between a– δ
and a+ δ, then we can be sure that f(x) lies
between L– εand L+ ε.
This says that no matter which level of precision
you care to choose (ε), all outputs of the function f(x)
for values xclose to a(namely, within a distance δof a)
will be within a distance εfrom L. Consider, for exam-
ple, the function f(x) = 5xfor values close to x= 2.
Notice that all outputs of the function are within a dis-
tance 0.1 from 10 if xis within a distance 0.02 from 2.
All outputs of the function are within a distance 0.001
from 10 if xis within a distance 0.0002 from 2. In fact,
for any small value ε, it is possible to describe a num-
ber δso that if xis within a distance δfrom 2, then
f(x)= 5xis within a distance εfrom 10. (In fact,
δ
=
will do.) This shows lim x→25x= 10.
If a function f(x) is continuous at x= a, then the
limit limx→af(x) exists and equals f(a).This, however,
need not always be the case (in which case we say that
fis discontinuous at a).
It is sometimes convenient to describe limit “just
from the left” or “just from the right.” Written as
limx→a–f(x),a limit from the left is defined as a value L
so that outputs of the function f(x) can be made as
close to Las we please by restricting xto values close
to and to the left of a(that is, for values of xbetween
a– δand a, for some number δ). A limit from the right,
written limx→a+f(x),is a value Lso that outputs of the
function f(x) can be made as close to Las we please by
restricting xto values close to and to the right of a
(that is, for values of xbetween aand a+ δ, for some
number δ). For example, in the graph above we have
limx→2–f(x) = 1 and limx→2+f(x) = 0. That the left and
right limits do not agree shows that the function is dis-
continuous at x= 2.
The word limit is also used in
INTEGRAL CALCULUS
in terms of a limit of integration. Given a definite inte-
gral ∫b
af(x)dx, the number ais called the lower limit of
integration, and bthe upper limit of integration.
See also
ASYMPTOTE
;
CONTINUOUS FUNCTION
;
DERIVATIVE
;
DIVERGENT
;
HISTORY OF CALCULUS
(essay);
LEFT DERIVATIVE
/
RIGHT DERIVATIVE
;
REMOVABLE DIS
-
CONTINUITY
; Z
ENO
’
S PARADOXES
.
limit from the left/right See
LIMIT
.
Lindemann, Carl Louis Ferdinand von (1852–1939)
German Number theory Born on April 12, 1852,
scholar Ferdinand von Lindemann is best remembered
for his 1882 proof that πis a
TRANSCENDENTAL NUM
-
BER
. This accomplishment finally settled the age-old
problem of
SQUARING THE CIRCLE
: by proving that πis
not a solution to a polynomial equation with integer
ε
–
5
1
–
ε
Lindemann, Carl Louis Ferdinand von 313
Left and right limits that do not match