Epsilon-delta definition of limits

A limit of a function f(x) at a given number a is the value that f(x) approaches as x approaches a, regardless of the value of x. This can be formally stated u...

A limit of a function f(x) at a given number a is the value that f(x) approaches as x approaches a, regardless of the value of x.

This can be formally stated using the epsilon-delta definition of a limit:

$\lim_{x \to a} f(x) = L \iff \forall \varepsilon > 0, \exists \delta > 0 \text{ such that } |x - a| < \delta \Rightarrow |f(x) - L| < \varepsilon$

In this definition:

$\varepsilon$ is the desired level of error in the limit.
$\delta$ is the allowed error in the calculation of the function value at x.
$0$ is the value of x approaching a.
$L$ is the value that $f(x)$ approaches as $x$ approaches $a$ .

The epsilon-delta definition provides a precise mathematical definition for the concept of a limit. It is a powerful tool for understanding and working with limits.

For example, let's consider the limit of the function f(x) = 1/x as x approaches 0. According to the epsilon-delta definition, this limit is equal to 1. This can be proven using the following steps:

$0 < \delta < \frac{1}{2}.$

If we take a look at the graph of f(x) = 1/x, we can see that the value of f(x) approaches 1 as x approaches 0. This verifies that the limit of f(x) = 1/x as x approaches 0 is equal to 1, according to the epsilon-delta definition