So far, we have proved three main results concerning limits and
arithmetical operations +,algebraic
nature of addition, multiplication and division, such
results are often grouped together under the title The Algebra Of
Limits
. I find this name very unfortunate for a number of reasons:
first, it suggests the work here has been algebraic (it hasn't: it has
been analytic); second, it suggests that the results are
true because of the special algebraic
nature of the
operations on limits being considered; third, the initials are "AOL".
To really understand what is going on here we need to take a slightly
more abstract view and look to see what addition, subtraction and
multiplication have in common. These are all functions, defined on the
real numbers, with two inputs
or arguments. If
Where, by this we mean: if the limits on the right hand side exist in the reals then the limit on the left exists and equals the expression given. This is the main content of the last two web pages on sums of sequences and products of sequences.
Other functions have this property such as the distance function
integer part
function,
pushed through
integer-part
signs. Examining the graph of jump
at
Definition.
Let
So for example, the function
Definition.
Let
Continuity is about pushing functions through limits
and the most important continuous functions are the arithmetic
ones such as plus and times. The following theorem covers the
most important cases of these:
Continuity of plus, times, etc.
The first four parts of this have already been proved. We sketch the proof of the others here.
Proposition.
The square root function
Proof.
Let _{
} as
Subproof.
So
Proposition.
The absolute value function
Proof.
These results can be combined to give other continuous functions.
The distance function
Continuity of the familiar functions we see here is useful in computing limits of what would otherwise be rather scary-looking sequences.
Example.
Let (_{ ) } be defined by
We compute its limit as
as
Example.
_{ +2 }=Let us suppose that _{
0
} as
But _{
+2
} so by the uniqueness
of limits we must have
The conclusion is that if the sequence (_{
)
}
converges to some
Pushing limits
through functions is a natural idea and can
save an enormous amount of time in calculations and proofs. You
have seen in this web page how it can be justified for some functions
and not for others. The functions for which we can manipulate limits
in this way are called continuous. Sometimes a function is
continuous at some points but not others. The integer-part function
and the division function are examples of these.
Implicitly assuming a function is continuous and
pushing a limit through
is one of the most common sources of errors in a mathematical
argument. There are many fallacious arguments
that prove
that 0=1 or some such absurdity that reply on assuming a
non-continuous function is continous to trick the reader. Please
always state in your own work reasons for such
arguments (and check the functions really are continuous).