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Limits - Algebra of Limits

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The limiting process respects addition, subtraction, multiplication and division as long as the limits and functions under consideration are well defined. In fact, below we formalise these as a theorem without proof.

theorem

Theorem: 
 Let f  and g be two functions such that both `lim_(x -> a)` f(x) and `lim_(x -> a)` g(x) exist.
Then

(i) Limit of sum of two functions is sum of the limits of the functions, i.e.,
`lim_(x -> a) [f(x) + g(x)]` = `lim_(x -> a) f(x) + lim _(x -> a) g(x)`.

(ii) Limit of difference of two functions is difference of the limits of the functions, i.e.,
`lim_(x -> a) [f(x) -g(x)] = lim_(x -> a) f(x) -lim _(x -> a) g(x)`.

(iii) Limit of product of two functions is product of the limits of the functions, i.e.,
`lim_(x -> a) [f(x)  . g(x)] = lim_(x -> a) f(x)  .  lim _(x -> a) g(x)`.

(iv) Limit of quotient of two functions is quotient of the limits of the functions (whenever the denominator is non zero), i.e., 
`lim_(x -> a) f(x)/g(x)  =(lim_(x->a) f(x))/(lim_(x-> a )  g(x))`

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