Algebra of Limits

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 x≤0

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

Example:

limx→1x2-1x-1

≠limx→1(x2-1)limx→1(x-1)

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.

summary

**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.

results

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

**Limit of Sum or Difference: ** Given that limx→af(x) and limx→ag(x) exists. Then

limx→a(f(x)±g(x))

=limx→af(x)±limx→ag(x)

limit of product is product of limits.

**Limit of Product: ** Given that limx→af(x) and limx→ag(x) exists. Then

limx→a(f(x)⋅g(x))

=limx→af(x)⋅limx→ag(x)

limit of quotient is quotient of limits.

**Limit of Quotient: ** Given that limx→af(x) and limx→ag(x) exists. Then

limx→a(f(x)g(x))

=limx→af(x)limx→ag(x)

limit of exponent is exponent of limit.

**Limit of Exponent: ** Given that limx→af(x) exists. Then

limx→a[f(x)n]

=[limx→af(x)]n

limit of root is root of limit.

**Limit of Root: ** Given that limx→af(x) and limx→ag(x) exists. Then

limx→a[f(x)1n]

=[limx→af(x)]1n

The variable in a limit can be changed.

Given

limx→0sinxx=1 ;

limx→0sin(x2)x

=limx→0xsin(x2)x2

=limx→0x×limy→0sinyy

where y=x2

by that definition, limx→0 changes to limy→0 .
=0×1

=0

Note: If, in another case, y=cosx then limx→0 changes to limy→1, as y=cos0=1.

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

limx→af(x)

=limy→g(a)f(g-1(y))

summary

**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 ∞-∞*

→ limx→a[f(x)±g(x)]=limx→af(x)±limx→ag(x)

**Limit of Product**

» Limit distributes over multiplication

*when value is not ∞×0*

→ limx→a[f(x)×g(x)]=limx→af(x)×limx→ag(x)

**Limit of Quotient**

» Limit distributes over division

*when value is not 0÷0 or ∞÷∞*

→ limx→a[f(x)÷g(x)]=limx→af(x)÷limx→ag(x)

**Limit of Exponent**

» Limit distributes over exponent

*when value is not ∞0 or 00*

→ limx→a[f(x)]n=[limx→af(x)]n

**Limit of Root**

» Limit distributes over root

*when value is not ∞0 or 00*

→ limx→a[f(x)]1n=[limx→af(x)]1n

**Change of Variable in a Limit**

» variable can be substituted

*when value is not any of the forms of 00*

→ limx→af(x) =limy→g(a)f(g-1(y))

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__