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.

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`.

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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