Formal Definition of Limits

Formal definition of limits: Let 'f' be any function defined on an open interval containing 'c'( not equal to c) and let 'L' be any real number, then
$\lim_{x->c}= L$

means that for each $\varepsilon$ >0 there exists $\delta$ >0 such that if
0 < | x - c| < $\delta$ then

|f(x) - L | < $\varepsilon$

Examples on Formal Definition of Limits

Find a $\delta $ for given $\varepsilon$
Example 1 :
Given : $\lim_{x->3}(2x - 5)$ = 1. Find $\delta$ such that
|(2x - 5) -1| < 0.01
whenever
0 < | x - 3| < $\delta $
Solution : As we know that |(2x - 5) -1| < 0.01 so the value of $\varepsilon$ = 0.01 To find the value of $\delta $ , we will make a connection between absolute values.
|(2x - 5) -1| and | x - 3|

|(2x - 5) -1| = |2x - 5 -1| = |2x - 6| = 2| x - 3|

2| x - 3| < 0.01

So, $\delta $ = $\frac{1}{2}(0.01)$

$\delta $ = 0.005

Example 2 : Find the limit $\lim_{x->4}(x+2)$ = L. Then use the $\varepsilon$ - $\delta $ definition to prove that the limit is L
Solution :
$\lim_{x->4}(x+2)$ = L

$\lim_{x->4}(x+2)$ = 4 + 2 = 6 = L

|(x+2) - 6| < $\varepsilon$

Since x close to 4
|x - 4| < $\delta $

|(x+2) - 6| = |x - 4|

$\varepsilon$ = $\delta $

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