LIMITS

 The concept of limit of a function is a fundamental concept in calculus .

 Lmit of a Function :

         Suppose x is a variable and 'a' is a constant . If x takes value closer and closer to 'a' but not equal to 'a' , then we say that x tends to 'a' . Symbolically it is denoted by x⟶ a .    

                          

we can observ that the value of x are very closed to a but not equal to a .When x > a and x takes values near a , foe example , x = a + 1/8 , x = a +1/4 ... etc, we say that x⟶a⁺  (x tends to a from larger value ). When x < a and x takes value near a , for example , x = a - 1/2 , a = a - 1/4 , x = a - 1/8 ..etc., then we sat that x⟶ a^- ( x tends to a from smaller value ).

Definition of Limits :

           Let f(x) is very near to l . This is expressed by | f(x) - l | < ε for any ε > 0 . Hence  ε can be arbitrarily small to ensure that f(x) is very near to l . If this condition is satisfied for all x near enough , then we can say that f(x) → l as x → a and a is expressed by 0 < |x - a| < δ where δ  > 0 .This δ  can be very small and depend upon f(x) and ε .

In short :  If given ε > 0 there exist  δ  > 0 such that  | f(x) - l | < ε for all   |x - a| < δ , then we say that  f(x) → l as x →  a  .

eg.,  Consider the example f(x) = 3x +1, take a = 0 and l = 1.

solution :  we want to fnd some δ  > 0  such that ,

 0 < |x - 0| < δ  ⇒ |(3x + 1 )-1| < ε 

if |3x| <  ε  i.e. if  3|x| < ε  i.e  if  |x| < ε /3

so we can chose δ < ε /3

∴ 0 < |x - 0| < δ ⇒ |f(x) - l | < ε 

∴ lim _{x →a} (3x + 1) = 1

One Sided Limit : 

   Lim _{x →a-} and Lim _{x →a+} ; if they exist are called one sided limit .

Left Hand Limit :  

   If given   ε > 0 there exist  δ  > 0 such that  | f(x) - l | < ε for all  x with a -  δ < x < a then                         lim _x →a- f(x) = 1.

Right Hand Limit :

   If given   ε > 0 there exist  δ  > 0 such that  | f(x) - l | < ε for all  x with a < x < a + δ  then                         lim _x →a+ f(x) = 1.   

Existanxe of a limit of a function at a oint x = a :

   If  lim _x →a+ f(x) =  lim _x →a- f(x)  = l , then limit of the function f(x) as x →a exist and its value is l. And if lim _x →a+ f(x) ≠  lim _x →a- f(x)  then  lim _x →a f(x)  does not exist .

Algebra of Limits :

   Let f(x) and g(x) be two different functions such that lim _x →a f(x) = l and  lim _x →a- f(x)  = m , then 

Note :

Method of Factorization :

   P(x) and Q(x) are polynomials in x such that  f(x) = P(x)/Q(x) . we consider lim _x →a f(x) .

Let's checklim _x →a p(x) and lim _x →a Q(x).

1. If lim _x →a Q(x) = m ≠ 0  , then  lim _x →a f(x) =  lim _x →a [P(x)/m] 
2. If  lim _x →a Q(x) = 0 , then (x-a)divides Q(x) . In such a case if (x-a) does not exist P(x) then  lim _x →a f(x) does not exist .
3. Further if lim _x →a p(x) is aslo 0 , then (x-a) is a factor of both P(x) anf Q(x) .

        So, lim _x →a f(x) = lim _x →a [ (P(x)/(x-a)) / (Q(x) / (x-a))] .

 Factorization of polynomial is a useful tool to dertmine the limits of rational algebric expressions.

Method of Rationalization :

      If the function in the limit involves a square root or a trignomwtric function , it may be possible to simplift the expression by multipling anf dividing by it's rationalizing factor.