Definition of Limit of a Function

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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 $x \to a$ if and only if, for every $ε > 0$ there exists $δ > 0$ such that

| f (x) - L | < ε,

whenever

0 < | x - a | < δ .

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 $f (x)$ has a limit $L$ at $x = a$ , if for every sequence ${x_{n}}$ , which has a limit at $a,$ the sequence $f (x_{n})$ has a limit $L .$ The Heine and Cauchy definitions of limit of a function are equivalent.

One-Sided Limits

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

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

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

lim_{x \to a} f (x) = lim_{x \to a - 0} f (x) = lim_{x \to a + 0} f (x) .

Solved Problems

Click or tap a problem to see the solution.

Example 1

Using the $ε - δ -$ definition of limit, show that $lim_{x \to 3} (3 x - 2) = 7.$

Example 2

Using the $ε - δ -$ definition of limit, show that $lim_{x \to 2} x^{2} = 4.$

Example 1.

Using the $ε - δ -$ definition of limit, show that $lim_{x \to 3} (3 x - 2) = 7.$

Solution.

Let $ε > 0$ be an arbitrary positive number. Choose $δ = \frac{ε}{3}$ . We see that if

0 < | x - 3 | < δ,

then

| f (x) - L | = | (3 x - 2) - 7 | = | 3 x - 9 | = 3 | x - 3 | < 3 δ = 3 \cdot \frac{ε}{3} = ε .

Thus, by Cauchy definition, the limit is proved.

Example 2.

Using the $ε - δ -$ definition of limit, show that $lim_{x \to 2} x^{2} = 4.$

Solution.

For convenience, we will suppose that $δ = 1,$ that is

| x - 2 | < 1.

Let $ε > 0$ be an arbitrary number. Then we can write the following inequality:

| x^{2} - 4 | < ε, \Rightarrow | x - 2 | | x + 2 | < ε, \Rightarrow | x - 2 | (x + 2) < ε .

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

5 | x - 2 | < ε (if | x - 2 | < 1), or | x - 2 | < \frac{ε}{2} .

Then for any $ε > 0$ we can choose the number $δ$ such that

δ = min (\frac{ε}{2}, 1) .

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