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 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