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