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

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THEORY OF LIMITS
13

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

has no meaning, being different from zero, for there exists no number such that if it be multiplied by zero, the product will equal .

Therefore division by zero is not an admissible operation.

Care should be taken not to divide by zero inadvertently. The following fallacy is an illustration.

The result is absurd, and is caused by the fact that we divided by .

15. Infinitesimals A variable whose limit is zero is called an infinitesimal.[1] This is written

, or, ,

and means that the successive numerical values of ultimately become and remain less than any positive number however small. Such a variable is said to become indefinitely small or to ultimately vanish.

If
, then ;

that 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.

16. The concept of infinity (). If a variable ultimately becomes and remains greater than any assigned positive number however large, we say increases without limit, and write

, or, .

If a variable ultimately becomes and remains algebraically less than any assigned negative number, we say decreases without limit, and write

, or, .

  1. Hence a constant, no matter how small it may be, is not an infinitesimal.