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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

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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
•  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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