C. Limit Theorems

We will now prove that a certain limit exists, namely the limit of f (x) = x as x approaches any value c. (That f(x) also approaches c should be obvious.)

THEOREM. If f (x) = x, then for any value c that we might name:

For, if a sequence of values of the variable x approaches c as a limit (Definition 2.1), then a sequence of values of the function f(x) = x will also approach c as a limit (Definition 2.2).

For example,

Theorems on limits

To help us calculate limits, it is possible to prove the following.

Let f and g be functions of a variable x. Then, if the following limits exist:

In other words:

1) The limit of a sum is equal to the sum of the limits.

2) The limit of a product is equal to the product of the limits.

3) The limit of a quotient is equal to the quotient of the limits,

3) provided the limit of the denominator is not 0.

Also, if c does not depend on x -- if c is a constant -- then

It should be clear from this example that to evaluate the limit of any power of x as x approaches any value, simply evaluate the power at that value.

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