Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/37

Rh zero, and the quotient is indeterminate; that is, any number whatever may be considered as the quotient, a result which is of no value.

$$\scriptstyle{\frac{a}{0}}$$ has no meaning, $$\scriptstyle{a}$$ being different from zero, for there exists no number such that if it be multiplied by zero, the product will equal $$\scriptstyle{a}$$.

Therefore division by zero is not an admissible operation.

A variable $$\scriptstyle{v}$$ whose limit is zero is called an infinitesimal. This is writtenand means that the successive numerical values of $$\scriptstyle{v}$$ ultimately become and remain less than any positive number however small. Such a variable is said to become indefinitely small or to ultimately vanish.

Ifthat is, the difference between a variable and its limit is an infinitesimal.

Conversely, if the difference between a variable and a constant is an infinitesimal, then the variable approaches the constant as a limit.

If a variable $$\scriptstyle{v}$$ ultimately becomes and remains greater than any assigned positive number however large, we say $$\scriptstyle{v}$$ increases without limit, and write

If a variable $$\scriptstyle{v}$$ ultimately becomes and remains algebraically less than any assigned negative number, we say $$\scriptstyle{v}$$ decreases without limit, and write