# Definition of Limit of a Function

## Cauchy and Heine Definitions of Limit

Let *f* (*x*) be a function that is defined on an open interval *X* containing *x* =
*a*. (The value *f* (*a*) need not be defined.)

The number *L* is called the limit of function *f* (*x*) as *x* → *a* if and only if, for every *ε* > 0 there exists *δ* > 0 such that

whenever

This definition is known as *ε−δ*- or Cauchy definition for limit.

There's also the Heine definition of the limit of a function, which states that a function \(f\left( x \right)\) has a limit \(L\) at \(x = a\), if for every sequence \(\left\{ {{x_n}} \right\}\) (with \(\left\{ {{x_n}} \right\}\) not equal to \(a\) for all \(n\) ), which has a limit at \(a,\) the sequence \(\left\{f\left( {{x_n}} \right)\right\}\) converges to \(L.\)

The Heine and Cauchy definitions of limit of a function are equivalent.

## One-Sided Limits

Let \(\lim\limits_{x \to a - 0} \) denote the limit as \(x\) goes toward \(a\) by taking on values of \(x\) such that \(x \lt a\). The corresponding limit \(\lim\limits_{x \to a - 0} f\left( x \right)\) is called the left-hand limit of \(f\left( x \right)\) at the point \(x = a\).

Similarly, let \(\lim\limits_{x \to a + 0} \) denote the limit as \(x\) goes toward \(a\) by taking on values of \(x\) such that \(x \gt a\). The corresponding limit \(\lim\limits_{x \to a + 0} f\left( x \right)\) is called the right-hand limit of \(f\left( x \right)\) at \(x = a\).

Note that the \(2\)-sided limit \(\lim\limits_{x \to a} f\left( x \right)\) exists only if both one-sided limits exist and are equal to each other, that is \(\lim\limits_{x \to a - 0}f\left( x \right) \) \(= \lim\limits_{x \to a + 0}f\left( x \right) \). In this case,

## Solved Problems

### Example 1.

Using the \(\varepsilon-\delta-\) definition of limit, show that \[\lim\limits_{x \to 3} \left( {3x - 2} \right) = 7.\]

Solution.

Let \(\varepsilon \gt 0\) be an arbitrary positive number. Choose \(\delta = {\frac{\varepsilon }{3}}\). We see that if

then

Thus, by Cauchy definition, the limit is proved.

### Example 2.

Using the \(\varepsilon-\delta-\) definition of limit, show that \[\lim\limits_{x \to 2} {x^2} = 4.\]

Solution.

For convenience, we will suppose that \(\delta = 1,\) that is

Let \(\varepsilon \gt 0\) be an arbitrary number. Then we can write the following inequality:

Since the maximum value of \(x\) is \(3\) (as we supposed above), we obtain

Then for any \(\varepsilon \gt 0\) we can choose the number \(\delta\) such that

As a result, the inequalities in the definition of limit will be satisfied. Therefore, the given limit is proved.