William Anthony Granville

“ If the given product contains factors, then since each infinitesimal may be assumed less than the th root of , the product can be made less than itself.
(4) If is a variable which approaches a limit different from zero, then the quotient of an infinitesimal by is also an infinitesimal. For if limit , and is any number numerically less than , then, by definition of a limit, will ultimately become and remain numerically greater than . Hence the quotient , where is an infinitesimal, will ultimately become and remain numerically less than , and is therefore by (2) an infinitesimal. ”

“ Theorem III. The limit of the quotient of two variables is equal to the quotient of the limits of the separate variables, provided the limit of the denominator is not zero.
Before proving these theorems it is necessary to establish the following properties of infinitesimals.
(1) The sum of a finite number of infinitesimals is an infinitesimal. ”

“ For instance, if we say that shall be the arc of smallest numerical value whose tangent is , that is, shall take on only values between and , then we are limited to the branch passing through the origin, and the condition for continuity is satisfied. (9) Similarly,
is found to be a many-valued function. Confining ourselves to one branch of the graph of
we see that as approaches zero from the left, approaches the limit and as approaches zero from the right, approaches the limit Hence the function is discontinuous when Its value for can be assigned at pleasure. ”