maths > calculus-limits

Algebra of Limits

    what you'll learn...

Understanding Algebra of Limits

 »  Finding limit of function as sub-expressions

    →  f(x)±g(x)f(x)±g(x)

    →  f(x)×g(x)f(x)×g(x)

    →  f(x)÷g(x)f(x)÷g(x)

    →  [f(x)]n[f(x)]n

    →  f(x)f(x) and y=g(x)y=g(x)

algebra is about operations + - * / ^

"Algebra of limits" means: Properties to find limit of functions given as algebraic operations of several functions.

Let us see this in detail.

The basic mathematical operations are
 •  addition and subtraction
 •  multiplication and division
 •  powers, roots, and logarithms.

Two or more function g(x) h(x) can form another function f(x).
f(x)=g(x)h(x)      where is one of the mathematical operations.

Will there be any relationship between the limits of the functions limg(x) ; limh(x) and the limit of the function limf(x)?
Algebra of limits analyses this and provides the required knowledge.

caution before using

In computing limit of a function, the value of the function or limit of the function changes :

 •  when a function evaluates to 0 in denominator

 •  When a function evaluates to

 •  at the discontinuous points of piecewise functions.

When applying algebra of limits to elements of a function, look out for the following cases.

 •  Expressions evaluating to 10 or 00 or ×0 or

    eg: 1x-1, x2-1x-1, tanxcotx, tanxsecx

 •  Expressions evaluating to - or +(-)
     eg: x2-4xx-1x
 •  discontinuous points of piecewise functions

     eg: {1  if  x>00  if  x0

The algebra of limit applies only when the above values do not occur.

The above is not applicable because it evaluates to 00.

Algebra of limits helps to simplify finding limit by applying the limit to sub-expressions of a function.
Algebra of limits may not be applicable to the sub-expressions evaluating to 0 or or at discontinuities.


Algebra of Limits: If a function f(x) consists of mathematical operations of sub-expressions f1(x), f2(x), etc. then the limit of the function can be applied to the sub-expressions.

If any of the sub-expressions or combination of them evaluate to 0 or then, the algebra of limit may not be applied to those sub-expressions.


limit of sum (or difference) is sum (or difference) of limits.

Limit of Sum or Difference: Given that limxaf(x) and limxag(x) exists. Then

limit of product is product of limits.

Limit of Product: Given that limxaf(x) and limxag(x) exists. Then

limit of quotient is quotient of limits.

Limit of Quotient: Given that limxaf(x) and limxag(x) exists. Then

limit of exponent is exponent of limit.

Limit of Exponent: Given that limxaf(x) exists. Then

limit of root is root of limit.

Limit of Root: Given that limxaf(x) and limxag(x) exists. Then

The variable in a limit can be changed.

limx0sinxx=1 ;
where y=x2
by that definition, limx0 changes to limy0 .     =0×1

Note: If, in another case, y=cosx then limx0 changes to limy1, as y=cos0=1.

Change of variable in a Limit: Given that y=g(x) exists at x=a. Then


Algebra of Limits

    →  If sub-expressions are not evaluating to 0 or then limit can be applied to sub-expressions.

    →  If sub-expressions are evaluating to 0 or , then look for the forms of 00.

Limit of Sum or Difference
 »  Limit distributes over Addition and Subtraction
when value is not -
    →  limxa[f(x)±g(x)]=limxaf(x)±limxag(x)

Limit of Product
 »  Limit distributes over multiplication
when value is not ×0
    →  limxa[f(x)×g(x)]=limxaf(x)×limxag(x)

Limit of Quotient
 »  Limit distributes over division
when value is not 0÷0 or ÷
    →  limxa[f(x)÷g(x)]=limxaf(x)÷limxag(x)

Limit of Exponent
 »  Limit distributes over exponent
when value is not 0 or 00
    →  limxa[f(x)]n=[limxaf(x)]n

Limit of Root
 »  Limit distributes over root
when value is not 0 or 00
    →  limxa[f(x)]1n=[limxaf(x)]1n

Change of Variable in a Limit
 »  variable can be substituted
when value is not any of the forms of 00
    →  limxaf(x) =limyg(a)f(g-1(y))


The outline of material to learn "limits (calculus)" is as follows.

Note : click here for detailed outline of Limits(Calculus).

    →   Indeterminate and Undefined

    →   Indeterminate value in Functions

    →   Expected Value

    →   Continuity

    →   Definition by Limits

    →   Geometrical Explanation for Limits

    →   Limit with Numerator and Denominator

    →   Limits of Ratios - Examples

    →   L'hospital Rule

    →   Examining a function

    →   Algebra of Limits

    →   Limit of a Polynomial

    →   Limit of Ratio of Zeros

    →   Limit of ratio of infinities

    →   limit of Binomial

    →   Limit of Non-algebraic Functions