Limit

Analysis Method Page 7 /30

LIMIT

1. LimitsofFunctionValues

Lets begin with an informal definition of limit, postponing the precise definition until weve gained more insight. Let (x) be defined on an open interval about 0x except possibly at 0x itself. If (x) gets arbitrarily close to L (as close to L as we like) for all x sufficiently close to 0x , we say that approaches the limit L as x approaches 0x and we write Lxf

xx= )(lim0 ,

which is read the limit of (x) as x approaches 0x is L. Essentially, the definition says that the values of (x) are close to the number L whenever x is close to 0x (on either side of ). This definition is informal because phrases like arbitrarily close and sufficiently close are imprecise; their meaning depends on the context.

2. DefinitionofLimit

Let (x) be defined on an open interval about except possibly at itself. We say that the limit of (x) as x approaches is the number L, and write

,)(lim0

Lxfxx

= if, for every number 0> , there exists a corresponding number 0> such that for all x,


(3) Product Rule: The limit of a product of two functions is the product of their limits.

( ) MLxgxfcx

=

)()(lim (4) Constant Multiple Rule:

The limit of a constant times a function is the constant times the limit of the function.

( ) Lkxfkcx

=

)(lim (5) Quotient Rule:

The limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero.

ML

xgxf

cx=

)()(lim 0M

(6) Power Rule:

If r and s are integers with no common factor and s 0, then ( ) srsr

cxLxf //)(lim =

provided that srL / is a real number. (If s is even, we assume that L > 0 )

The limit of a rational power of a function is that power of the limit of the function, provided the latter is a real number.

4. LimitsofPolynomials

Limits of polynomials can be found by substitution

5. LimitsofRationalFunctions

Limits of rational functions can be found by substitution if the limit of the denominator is not zero

6. TheSandwichTheorem

Suppose that )()()( xhxgxf for all x in some open interval containing c, except possibly at cx = itself. Suppose also that

Lxhxfcxcx

== )(lim)(lim Then Lxg

cx= )(lim


7. OneSidedLimits

To have a limit L as x approaches c, a function must be defined on both sides of c and its values (x) must approach L as x approaches c from either side. Because of this, ordinary limits are called two-sided.

If fails to have a two-sided limit at c, it may still have a one-sided limit, that is, a limit if the approach is only from one side. If the approach is from the right, the limit is a right-hand limit. From the left, it is a left-hand limit.

Intuitively, if (x) is defined on an interval (c, b), where c < b and approaches arbitrarily close to L as x approaches c from within that interval, then has right-hand limit L at c. We write Lxf

cx=+ )(lim .

The symbol + cx means that we consider only values of x greater than c.

Similarly, if (x) is defined on an interval (a, c), where and approaches arbitrarily close to M as x approaches c from within that interval, then has left-hand limit M at c. We write Mxf

cx= )(lim .

The symbol cx means that we consider only x values less than c.

Fig 3 Different right-hand and left-hand limits at the origin.

A function (x) has a limit as x approaches c if and only if it has left-hand and right-hand limits there and these one-sided limits are equal: Lxf

cx= )(lim Lxfcx =+ )(lim and Lxfcx = )(lim .

8. DefinitionsofRightHandandLeftHandLimits

We say that (x) has right-hand limit L at x0 and write Lxf

xx=+ )(lim0 (See Figure )

if for every number 0> there exists a corresponding number 0> such that for all x +


Fig 4 Intervals associated with the definition of right-hand limit.

We say that has left-hand limit L at and write Lxf

xx= )(lim0 (See Figure )

if for every number 0> there exists a corresponding number 0> such that for all x 00 xxx


Lxfx

=

)(lim

if, for every number there exists a corresponding number N such that for all x Nx < 0) (4) 0lim =

n

nx ( 1

Documents

Limit