# Limits of functions*Limit of a function. Some remarkable limits. * Infinitesimal and infinite values. Finite limit. Infinite limit. Notion of an infinity. *Limit of a function. **A number*** **L is called a limit of a function y = f (* x *) *as x tends a :*
*if and only if for any* > 0 *one can find such a positive number* = ( ), *depending on* , * that* | *x* – *a* | < *implies that* | *f *( *x* ) – *L* | < .This definition means, that *L *is a *limit* of a function *y *= *f *( *x *) , if the function value approaches unrestrictedly to *L* , when the argument value * x * approaches *a*. Geometrically it means, that for any > 0 it is possible to find such a number , that if *x* is within an interval ( *a* – , *a* + ), then a function value is within an interval ( *L* – , *L* + ). Note, that according to this definition, a function argument *x* only __ approaches__ *a*, not adopting this value! It must be considered at calculating limit of any function at a point of its discontinuity ( i.e., where this function doesn’t exist ). E x a m p l e . Find :
S o l u t i o n . Substituting *x* = 3 into the expression we’ll receive a meaningless expression ( see "About meaningless expressions" in the section "Powers and roots" of the part "Algebra"). Therefore we’ll solve in a different way: Here the fraction canceling is valid, because *x* 3 , it only __approaches__ 3. Now we have:
because, if *x* approaches 3 , then *x* + 3 approaches 6 . *Some remarkable limits. *
*Infinitesimal and infinite value. *If limit of some variable is equal to 0, this variable is called an *infinitesimal*. E x a m p l e . The function *y * = is an infinitesimal, if *x* approaches 4, because If an absolute value of some variable increases unboundedly, then this variable is called an *infinite value*. An infinite value has no a *finite* limit, but it has so called an * infinite* limit; this fact is written as:
The symbol ("infinity") doesn’t mean some number, it means only that the fraction increases unboundedly if *x* approaches 3. It should be noted, that the fraction can be both positive (at *x* > 3) and negative (at *x* < 3). If an infinite value can be only positive at any values of *x*, it is marked in a record. For example, at *x* 0 the function *y* = *x*^{–2} is an infinite value, but it is positive both at *x* > 0 and at *x* < 0 ; this is expressed as: On the contrary, the function *y* = – * x *^{–2} is always negative, therefore According to this, a result of our example can be written as: then this fraction approaches 1 , i.e. its limit is 1. Back |