ISC (Arts) Class 11CISCE
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Limits - Limits of Polynomials and Rational Functions

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A function f is said to be a polynomial function of degree n f(x) = `a_0+ a_1x+a_2x^2+ . . + a_nx^n`  , where a_1s are real numbers such that `a _n`  ≠ 0 for some natural number n. 
`lim_(x->a)` x = a .
Hence 
`lim_(x -> a)x^2 = lim_(x->a) (x.x)` = `lim_(x->a) x . lim_(x->a) x = a.a =a^2`

An easy exercise in induction on n tells us that
`lim_(x-> a) x^n = a^n`
Now, let f(x) = `a_0 + a_1x + a_2x^2 + ...+a_nx^n` be a polynomial function.
Suppose of each of `a_0 , a_1x , a_2x^2 , ...., a_nx^n ` as a function , we have  

`lim_(x ->a) f(x) = lim_(x -> a) [a_0 + a_1x + a_2 x^2 + ...+a_nx^n]`

= `lim_(x -> a) a_0 + lim_(x -> a) a_1x  + lim_(x -> a) a_2x^2 + ... + a_nx^n`

= `a_0 +  a_1 lim_(x ->a) x + a_2 lim_(x ->a) x^2 + ... + a_n lim_(x ->a) x^n.`

= `a_0 + a_1a + a_2a^2 + ... + a_na^n`

= f(a)
A function f is said to be a rational function, if f(x) = `g(x)/(h(x))` ,  where g(x) and h(x) are polynomials such that h(x) ≠ 0.
Then `lim_(x ->a) f(x) = lim_(x ->a)g(x)/(h(x)) =(lim_(x -> a) g(x))/(lim_(x ->a) h(x)) = g(a)/(h(a))`.

Case 1 - h(a) = 0  and g(a) = k

`g(a)/(h(a)) = k/0 = ∞ `
Limit does not exist (undefined)
Example - 
`lim_(x->2) (x^3 - 2 )/(x - 2) = (2^3 - 2)/(2-2) = (8-2)/0 = 6/0 = ∞`

Case 2 -  
h(a) = 0 and  g(a) = 0

`g(a)/(h(a)) = 0/0`
Example - `lim_(x->2) (x^2 - 4)/(x-2) = (2^2 - 4)/(2 - 2) = (4 - 4)/(2 - 2) = 0/0`

Case 3 -  h(a) = k  and g(a) = 0
`g(a)/(h(a)) = 0/k = 0`
Example - `lim_(x->2) (x - 2)/(2x + 2) = (2 - 2)/(2 . 2 + 2) = 0/(4 + 2) = 0/6 = 0`

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