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Limit of Polynomials


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Limit of Polynomials

Apply Algebra of Limits
For a function f(x)=anxn+an-1xn-1++a1x1+a0

limxaf(x)

    =anlimxaxn+an-1limxaxn-1++a1limxax1+a0

limit of a polynomial

Using the algebra of limits, the limit of function f(x)=3x3+2x2-1 at x=-2 is found by applying limit to the three terms of the function.

limit of function f(x)=3x3+2x2-1 at x=-2

By Substitution :
f(x)x=-2
    =3(-2)3+2(-2)2-1
    =3(-8)+2(4)-1
    =-17

left-hand-limit :
limx-2-f(x)
    =3(-2-δ)3+2(-2-δ)2-1
substitute δ=0
    =3(-8)+2(4)-1
    =-17

right-hand-limit :
limx-2+f(x)
    =3(-2+δ)3+2(-2+δ)2-1
substitute δ=0
    =3(-8)+2(4)-1
    =-17

Summary :
limits of function f(x)=3x3+2x2-1 at x=-2

f(x)x=-2=-17

limx-2-f(x)=-17

limx-2+f(x)=-17

All three values are equal. So the function is continuous.

summary

Limit of a polynomial: For a function f(x)=anxn+an-1xn-1++a1x1+a0
limxaf(x)
    =anlimxaxn+an-1limxaxn-1+
        +a1limxax1+a0

Outline

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