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mathsLimit of a functionAlgebra of Limits

Algebra of Limits : Introduction

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

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What does "Algebra of limits" mean?

  • Properties to find limit of functions given as algebraic operations of several functions
  • 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
  • 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.

`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`.

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.

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.

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