Limit Continuity and Infinity

To analysis of real number and real function sometimes we get a function like bellow

f(x) = (x - 3)/(x^2 - 9) . If we ask what will be the value at x = 3 . Ans is NaN or undefined .

Now take a close look to this function .

let's calculate f(x + δ) where δ is a positive number small than the smallest number (we can think) .
This been represented as δ → 0 .

Now see what happen if we minimize and minimize δ .

f(x + δ)           δ
0.3225          0.1
0.3322          0.01
0.3332          0.001
0.3333          0.0001

So we see that the value is coming closure to 0.3333... ~ 1/3 as we minimize δ .



This value where the number tends to by decreasing the value of δ is called the limit (here right limit as we did x + δ ) . 

So in notation  Lt  (δ  → 0) ( f(x + δ ) ) = 1/3 .Similarly we can find left limit by doing f(x - δ ) and same procedure .

If left limit and right limit is same then we tell that    Lt (x  →  3 ) f(x) exist and = 1/3 .

Formal Definition of limit : We can say that  Lt (x  →  a ) f(x) exist and = l iff  for (δ  → 0)   0 <| x - a | < δ  
∃ e  → 0 s.t .  | f(x + a) - l | <  e ( right limit )  and  | f(x - a) - l | <  e ( left limit ) .


Continuity : We saw for the previous function that f(3) does not exist . Now if f(3) exist and 
 Lt (x  →  3 ) f(x) = f(3) then we would say that the function is continuous on that point .


Infinity is a concept which is the number greater than any number other than it .