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Limits involving Binomial Expressions

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Limits involving Binomial Expressions

 »  Special case of canceling factors in numerator and denominator
    →  limxaxn-anx-a=nan-1limxaxnanxa=nan1

 »  With change of variable x=1+yx=1+y and constant a=1a=1
    →  limy0(1+y)n-1y=nlimy0(1+y)n1y=n


Factoring binomials refers to factoring of an-bnanbn.

The word binomial means: of two terms.

The expression an-bnanbn is a two variate binomial of degree n.
two variate : a and b are two variables
binomial : an and -bn are the two terms
degree n: The maximum power is n

Factoring the binomials is given by


The value of f(x)=x4-16x-2 at x=2 is 00 by direct substitution x=2.

The limit of f(x)=x4-16x-2 at x=2 is 4×23.

limit of f(x)=x4-16x-2 at x=2.
On substitution of x=2 the function evaluates to 00

Limit of the function is

note: (x-2) is not canceled at x=2. But, limit is applied to x-2x-2

To find limit of functions with binomials, factor the binomials to cancel out the factor involving 0.

What is the limit for f(x)=x15-1x10-1 at x=1
The answer is '32'


Limit of expressions with Binomials: For any positive integer n


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