The epsilon-delta definition of limit says that if limx→a f(x) = L, then for every ε>0, there exists a δ>0 such that |f(x)-L| < ε whenever 0<|x-a|<δ.
In this post, we will learn the epsilon-delta definition of a limit with examples. We also provide the negative statement of the epsilon-delta definition of limits.
Epsilon-Delta Definition of limit
Let f(x) be a function of a variable x such that f(x) tends to L when x tends to a, that is,
$\lim\limits_{x \to a} f(x) = L$.
The epsilon-delta definition of this limit is given as follows: for every ε>0, there exists a δ>0 such that
|f(x)-L| < ε whenever 0<|x-a|<δ
holds true.
Let us understand the above ε-δ definition with the help of an example.
Example: Show that $\lim\limits_{x \to 0} x^2 = 0$ by epsilon-delta method.
Solution:
Let ε>0 be a given positive number. we need to find a δ>0 such that
$|x^2-0| < \epsilon$ whenever 0<|x-0|<δ …(I)
Choose δ = $\sqrt{\epsilon}$. Then δ>0. Now, for 0< |x| < δ = $\sqrt{\epsilon}$ we have
|x| < $\sqrt{\epsilon}$
$\Rightarrow |x|^2 < \epsilon$
$\Rightarrow |x^2| < \epsilon$ whenever 0 < |x-0| < δ =$\sqrt{\epsilon}$.
Thus (I) is true. Hence by epsilon-delta definition of limit, we can conclude that limx→0 x2 = 0.
We will now give the negative statement of the epsilon-delta definition of limit.
Negation of Epsilon Delta Definition of limit
Suppose that $\lim\limits_{x \to a}$ f(x) = L, that is f(x) tends \to L when x→a. The negation of the epsilon-delta definition says the following:
f(x) does not tend \to L when x→a then it is not true that for every ε>0, there exists a δ>0 such that
|f(x)-L| < ε, whenever 0<|x-a|<δ.
That is, if f(x) does not tend to L when x→a, then there exists some ε>0 such that for every δ>0 there exists some point x in the set {x : 0<|x-a|<δ} such that
|f(x)-L| ≥ ε.
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