Before we move on to the evaluation of other limits, we prove the following useful result:
Theorem:
Suppose is continuous at
. Suppose
, when I is an open interval and
. If
exists and is equal to c, then
.
Proof.
Since f is continuous at , for each
, we can find
such that
for
.
Since , for the given
, we can find a
such that
for
.
So, for , we have
since
. QED.
Corollary.
Let I and J be open intervals, be continuous at
. If
is continuous at
, then
is continuous at
. This means that the composition of continuous functions is continuous.
Exercise.
If f is continuous, show that is continuous. (where |f|(x)=|f(x)|).
Some Useful Limits.
(1)(a) for a positive integer n. For this, note that
for
, and hence,
.
(1)(b) for a negative integer n if
.
If n is a negative integer write , where
.
(1)(c) , for
and when
.
Write ,
. So we have have
,
.
which in turn equals
(2) where
is measured in radians.
In some cases, when the function is defined and there is an l such that for every
we have an
such that
for
we say that tends to l as x tends to
.We write
Be warned that is not a real number to which x is coming close. Similarly, we may define
. If for every
we can find a
such that
for
, then we can write
. A similar definition for
can be given.
(3)(a) (3b)
Proof.
(3a) Let so that
giving us
. This implies that
Now, as , n also tends to
and in such a case both the right hand and left hand side of the above inequality tend to e. So,
. QED.
(3b) In the case of , let us write
so that
Now, . So, we have
. QED.
Corollary.
.
The above is an exercise for you.
Corollary.
.
The above is an exercise for you.
Corollary.
.
It will be appropriate to mention something about left continuity and right continuity of functions.
Definition.
If , then we say that the function f is left continuous at
. If
, then we say that the function is right continuous at
.
Thus, a function may be left continuous but not right continuous, in which case the function cannot be continuous. It may be right continuous but not left continuous, in which case also the function cannot be continuous. If a function is both left continuous and right continuous, only then is it continuous.
More later,
Nalin Pithwa