life tables 311

As another example we have:

cos (0) = 1.

If we make the assumption that the derivative g(x)

is not zero at x= a, then the proof of l’Hôpital’s rule is

relatively straightforward:

Write the limit as .

Since f(a) = g(a) = 0, we have

The result is still valid if g′(a) = 0, but one must make

clever use of the

MEAN

-

VALUE THEOREM

to establish it.

Johann Bernoulli is the one who discovered and proved

l’Hôpital’s rule.

It is permissible to apply l’Hôpital’s rule to the

same limit more than once, that is, to differentiate the

numerator and denominators each a number of times.

To illustrate, we have:

The theorem also holds for limits as x→∞.

The rule also works for a ratio of functions

f(x)/g(x) if each function separately tends to infinity as

x→a, (that is, we have an indeterminate ratio ∞/∞).

One notes that the functions 1/f(x) and 1/g(x) each

tend to zero as x→a, and so:

by l’Hôpital’s rule. Consequently:

from which it follows that . As

examples, we have:

and

liar’s paradox The sixth century

B

.

C

.

E

. Cretan

prophet Epimenides is purported to have said, “All Cre-

tans are liars,” a statement now referred to as the liar’s

paradox. It is difficult to determine whether Epi-

menides, himself a Cretan, is telling the truth here. The

statement, however, is not a true

PARADOX

. It may be

the case that Epimenides is aware of at least one honest

countryman and is making here a false statement via the

use of the word all. A sharper version of the intended

paradox lies in the statement, “This sentence is false.”

One may also consider the logical consequences

of variations of this statement. For example, what if

Epimenides had stated instead: “All Cretans are truth-

tellers”?

See also

SELF

-

REFERENCE

.

life tables (mortality tables) Based on census results

and results from medical and social studies, life tables

are tables of values indicating the proportion of people

of a certain age expected to live to successively higher

ages. Different tables are provided for different specific

populations and ages (for example, Australian males

age 40, Canadian females age 25) and are used exten-

sively by insurance companies to analyze risk in issuing

insurance policies, as well as in scientific studies. For

example, such tables can be used to compare mortality