Calculus

Limits and Continuity of Functions

Limits and Continuity Logo

Definition of Limit of a Function

  • Cauchy and Heine Definitions of Limit

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

    The number \(L\) is called the limit of function \(f\left( x \right)\) as \(x \to a\) if and only if, for every \(\varepsilon \gt 0\) there exists \(\delta \gt 0\) such that

    \[\left| {f\left( x \right) – L} \right| \lt \varepsilon ,\]

    whenever

    \[0 \lt \left| {x – a} \right| \lt \delta .\]

    This definition is known as \(\varepsilon-\delta-\) 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 \(\left\{ {{x_n}} \right\}\), which has a limit at \(a,\) the sequence \(f\left( {{x_n}} \right)\) has a limit \(L.\) 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,

    \[{\lim\limits_{x \to a}f\left( x \right) = \lim\limits_{x \to a – 0}f\left( x \right)} ={ \lim\limits_{x \to a + 0}f\left( x \right).}\]


  • Solved Problems

    Click a problem to see the solution.

    Example 1

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

    Example 2

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

    Example 3

    Using the \(\varepsilon-\delta-\) definition of limit, find the number \(\delta\) that corresponds to the \(\varepsilon\) given with the following limit:
    \[{\lim\limits_{x \to 7} \sqrt {x + 2} = 3,\;\;\;}\kern-0.3pt{\varepsilon = 0.2}\]

    Example 4

    Prove that \(\lim\limits_{x \to \infty } {\large\frac{{x + 1}}{x}\normalsize} = 1\).

    Example 5

    Prove that \(\lim\limits_{x \to \infty } {\large\frac{{2x – 3}}{{x + 1}}\normalsize} = 2\).

    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 \(\delta = {\large\frac{\varepsilon }{3}\normalsize}\). We see that if

    \[0 \lt \left| {x – 3} \right| \lt \delta, \]

    then

    \[{\left| {f\left( x \right) – L} \right| = \left| {\left( {3x – 2} \right) – 7} \right|} ={ \left| {3x – 9} \right| }
    ={ 3\left| {x – 3} \right| \lt 3\delta } = {3 \cdot \frac{\varepsilon }{3} = \varepsilon .}
    \]

    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 \(\delta = 1,\) i.e.

    \[\left| {x – 2} \right| \lt 1.\]

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

    \[{\left| {{x^2} – 4} \right| \lt \varepsilon ,\;\;}\Rightarrow
    {\left| {x – 2} \right|\left| {x + 2} \right| \lt \varepsilon ,\;\;}\Rightarrow
    {\left| {x – 2} \right|\left( {x + 2} \right) \lt \varepsilon .}
    \]

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

    \[{5\left| {x – 2} \right| \lt \varepsilon \;\;(\text{if } \left| {x – 2} \right| \lt 1),\;\;}\kern-0.3pt
    {\text{or}\;\left| {x – 2} \right| \lt \frac{\varepsilon }{2}.}
    \]

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

    \[\delta = \min \left( {\frac{\varepsilon }{2},1} \right).\]

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

    Page 1
    Problems 1-2
    Page 2
    Problems 3-5