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Algebra of Limits

Algebra of Limits

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Understanding Algebra of Limits


 »  Finding limit of function as sub-expressions

    →  `f(x) +- g(x)`

    →  `f(x) xx g(x)`

    →  `f(x) -: g(x)`

    →  `[f(x)]^n`

    →  `f(x)` and `y=g(x)`


 »  Algebra of Limits

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

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

Algebra of Limits : Introduction

plain and simple summary

nub

plain and simple summary

nub

dummy

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 `oo` or at discontinuities.

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This topic explains the observations one has to take before applying algebra of limits.


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Starting on learning "Introduction to Algebra of Limits". ;; This topic explains the observations one has to take before applying algebra of limits.

What does "Algebra of limits" mean?

  • Properties to find limit of functions given as algebraic operations of several functions
  • Applying limit in practical applications

The answer is 'Properties to find limit of functions given as algebraic operations of several functions'

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

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

Will there be any relationship between the limits of the functions `lim g(x)` ; `lim h(x)` and the limit of the function `lim f(x)`?
Algebra of limits analyses this and provides the required knowledge.

In computing limit of a function, when does value of the function or limit of the function change?

  • when a function evaluates to `0` in denominator
  • When a function evaluates to `oo`
  • at the discontinuous points of piecewise functions
  • all the above

The answer is 'all the above'

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

 •  Expressions evaluating to `1/0` or `0/0` or `oo xx 0` or `oo / oo`

    eg: `1/(x-1)`, `(x^2-1)/(x-1)`, `tan x cot x`, `(tan x)/(sec x)`

 •  Expressions evaluating to `oo - oo` or `oo + (-oo)`
     eg: `(x^2-4x)/x - 1/x`
 •  discontinuous points of piecewise functions

     eg: `{(1, quad if quad x>0),(0, quad if quad x<=0) :} `

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

Example:
`lim_(x->1) color(deepskyblue)(x^2-1)/color(coral)(x-1)`
`quad quad != color(deepskyblue)(lim_(x->1) (x^2-1))/color(coral)(lim_(x->1)(x-1) )`
The above is not applicable because it evaluates to `0/0`.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Algebra of Limits: If a function `f(x)` consists of mathematical operations of sub-expressions `f_1(x)`, `f_2(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 `oo` then, the algebra of limit may not be applied to those sub-expressions.



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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What does "Algebra of limits" mean?
find;given;algebraic;several
Properties to find limit of functions given as algebraic operations of several functions
practical;applications;applying
Applying limit in practical applications
The answer is 'Properties to find limit of functions given as algebraic operations of several functions'
The basic mathematical operations are ;; addition and subtraction ;; multiplication and division ;; powers and roots. ;; Two or more function g of x , h of x can form another function f of x .;; f of x = g of x star h of x ; where star is one of the mathematical operations. ;; Will there be any relationship between the limits of the functions limit g of x ; limit h of x and the limit of the function limit f of x ? ;; Algebra of limits analyses this and provides the required knowledge.
In computing limit of a function, when does value of the function or limit of the function change?
0;denominator
when a function evaluates to 0 in denominator
infinity
When a function evaluates to infinity
discontinuous;piecewise;points
at the discontinuous points of piecewise functions
all;above
all the above
The answer is 'all the above'
When applying algebra of limits to elements of a function, look out for the following cases. ;;Expressions evaluating to 1 by 0, or 0 by 0, or infinity into 0, or infinity by infinity ;;for example: 1 by x minus 1, x squared minus 1 by x minus 1 , tan x cot x , tan x by sec x ;; Expressions evaluating to infinity minus infinity, or infinity plus minus infinity ;; for example: x squared minus 4x by x, minus 1 by x ;; discontinuous points of piecewise functions;; for example : 1 if x greater than 0, 0 if x less than equal to 0 ;; The algebra of limit applies only when the above values do not occur. ;; Example: limit x tending to 1 x squared minus 1 by x minus 1;; is not equal to, limit x tending to 1, x squared minus 1 as numerator divided by, limit x tending to 1, x minus 1 as denominator;; The above is not applicable because it evaluates to 0 by 0.
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 infinity, or at discontinuities.
Algebra of Limits: If a function f of x consists of mathematical operations of sub-expressions f 1 of x , f 2 of 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 infinity then, the algebra of limit may not be applied to those sub-expressions.

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