Basic theorems of limits

Basic theorems of limits

1-The limit of the algebraic sum of a finite number of variables is equal to the sum of their limits.

For the sake of exactness let us take the algebraic sum x-y+z of three simultaneously varying magnitudes. We suppose that x, y and z tend respectively to limits a, b and c. We show that the sum tends to the limit a - b + c.

We have:

where  are infinitesimals. We can write for the sum:

The first bracket on the right-hand side of this equation is a constant, and the second is an infinitesimal. Hence:

2-The limit of the product of a finite number of variables is equal to the product of their limits.

We confine ourselves to the case of the product x*y of two variables. We suppose that x and y vary simultaneously, tending respectively to limits a and b, and we show that x*y tends to the limit a*b.

We have by hypothesis:

where  are infinitesimals; hence:

Using both of the properties of infinitesimals, we see that the sum in the bracket on the right of this equation is an infinitesimal, and hence we have:

3-The limit of a quotient is equal to the quotient of the limits, provided the limit of the denominator is not zero.

We take the quotient x/y, and suppose that x and y tend simultaneously to their respective limits a and b, where  We show that x/y tends to a/b.

To prove the theorem, it is sufficient to show that the difference a/b - x/y is an infinitesimal. By hypothesis:

where  are infinitesimals. Hence:

The denominator of the fraction on the right of this equation is the product of two factors, and tends to b^2. Thus, from some initial moment of its variation, it is greater than b^2/2, the fraction as a whole being included between zero and 2/b^2, i.e. the fraction is bounded. The term  is an infinitesimal. Hence, the difference a/b-x/y is an infinitesimal, and

the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek