Theorem:

The limit of the constant function , where c is a constant, at any value of x is equal to c. That is, .

Proof:

The statement is true for all values of .

So we can say that , no matter what values we choose for and a. So, .

Theorem:

The limit of the identity function at some point is equal to the value of x at that point, or, in symbols:

.

Proof:

We want to find some positive for each positive such that the statement , or is true.

We can clearly put and it is easy to complete the proof.

Lemma:

If and , then .

Proof:

For each we want to find such that

.

Since , and 1 is a positive number, there is some such that .

Using the properties of absolute values, we have . So,

. (1)

We also have , but is positive. So, there exists a number such that

(2)

From (1) and (2) and by taking we have

.

Thus, .

Theorem:

If we have two functions f and g whose limits exist at a number a, then,

.

Proof:

Let and . Suppose we have a positive number . Then is also positive. Since , there is a positive number such that

. (1)

Similarly, there exists a positive number such that

(2)

Putting , then from (1) and (2) we have

Thus, .

Theorem:

if and only if .

Proof:

Assume that . We will have

.

On the other hand, if , then

.

Theorem:

If we have two functions f and g whose limits exist at a number a, then,

.

Suppose that and . Then,

and .

Now, . Therefore, by taking the limits of both sides, we obtain

Lemma:

If , then .

Theorem:

If and , then .

Proof:

.

Theorem:

If n is any positive integer, then , where a is positive when n is even, otherwise it can be any real number.

Theorem:

If , there is an interval , such that for all .

Similarly, if , there is an interval , such that for all .

Theorem:

Suppose we have three functions f, g and h such that for all x in an open interval I except possibly at one point . If , then .

Theorem:

Consider two constants A and B. If for all , then A=B.