LIMIT OF FUNCTIONS OF REAL VARIABLE
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.
Let f be a function define for values of x near the point xo. Then the real number L is said to be the limit of f(x) as x approaches xo if, for each E>0 there exists a δ >0 such that |f(x) – L| < E, whenever 0 < |x – x0|< δ ; and we write
If f(x) = x2, show that
We have to show that given E > 0 there is a δ > 0 ( in general, depending upon E) such that
|x2 – 9| < E, when 0 < | x – x0 | < δ . Let us choose δ ≤ 1 so that 0< | x-3 | < δ ≤ 1. Then we have |x2 – 9| = |(x – 3) (x + 3)| = |x – 3||x + 3| < δ |x+ 3| ≤ δ( |x – 3|+ 6) < 7δ .
Take δ = min (1, E/2) , the smaller of the two.
Then, we have |x2 – 9|< E, whenever 0 < |x – 3|< δ ≤ 1
We state without proof, the following easy prepositions.
If f and g are functions define for value of x near xo, and if