Calculus

Limits and Continuity of Functions

Limits and Continuity Logo

The Number e

  • The number \(e\) is defined by:

    \[e = \lim\limits_{n \to \infty } {\left( {1 + \frac{1}{n}} \right)^n}.\]

    The number \(e\) is a transcendental number which is approximately equal to \(2.718281828\ldots\) The substitution \(u = {\large\frac{1}{n}\normalsize}\) where \(u = {\large\frac{1}{n}\normalsize} \to 0\) as \(n \to \pm \infty,\) leads to another definition for \(e:\)

    \[e = \lim\limits_{u \to 0} {\left( {1 + u} \right)^u}.\]

    Here we meet with power expressions, in which the base and power approach to a certain number \(a\) (or to infinity). In many cases such types of limits can be calculated by taking logarithm of the function.


    Solved Problems

    Click a problem to see the solution.

    Example 1

    Calculate the limit \(\lim\limits_{n \to \infty } {\left( {1 + {\large\frac{1}{n}}\normalsize} \right)^{n + 5}}.\)

    Example 2

    Find the limit \(\lim\limits_{x \to \infty } {\left( {1 + {\large\frac{1}{x}}\normalsize} \right)^{3x}}.\)

    Example 3

    Calculate the limit \(\lim\limits_{x \to \infty } {\left( {1 + {\large\frac{6}{x}}\normalsize} \right)^x}.\)

    Example 4

    Find the limit \(\lim\limits_{x \to 0} \sqrt[\large x\normalsize]{{1 + 3x}}.\)

    Example 5

    Find the limit \(\lim\limits_{x \to \infty } {\left( {\large\frac{{x + a}}{{x – a}}\normalsize} \right)^x}.\)

    Example 6

    Calculate the limit \(\lim\limits_{x \to \infty } {\left( {\large\frac{x}{{x + 1}}\normalsize} \right)^x}.\)

    Example 7

    Evaluate the limit \(\lim\limits_{x \to \infty } {\left( {\large\frac{{x + 3}}{{x – 2}}\normalsize} \right)^{x – 1}}.\)

    Example 8

    Find the limit \(\lim\limits_{x \to a} {\large\frac{{\ln x – \ln a}}{{x – a}}\normalsize},\) \(\left( {a \gt 0} \right).\)

    Example 9

    Calculate the limit \(\lim\limits_{x \to 0} {\left( {1 + \sin x} \right)^{\large\frac{1}{x}\normalsize}}.\)

    Example 1.

    Calculate the limit \(\lim\limits_{n \to \infty } {\left( {1 + {\large\frac{1}{n}}\normalsize} \right)^{n + 5}}.\)

    Solution.

    \[
    {\lim\limits_{n \to \infty } {\left( {1 + \frac{1}{n}} \right)^{n + 5}} }
    = {\lim\limits_{n \to \infty } \left[ {{{\left( {1 + \frac{1}{n}} \right)}^n}{{\left( {1 + \frac{1}{n}} \right)}^5}} \right] }
    = {\lim\limits_{n \to \infty } {\left( {1 + \frac{1}{n}} \right)^n} \cdot \lim\limits_{n \to \infty } {\left( {1 + \frac{1}{n}} \right)^5} }
    = {e \cdot 1 = e.}
    \]

    Example 2.

    Find the limit \(\lim\limits_{x \to \infty } {\left( {1 + {\large\frac{1}{x}}\normalsize} \right)^{3x}}.\)

    Solution.

    By the product rule for limits, we obtain

    \[
    {\lim\limits_{x \to \infty } {\left( {1 + \frac{1}{x}} \right)^{3x}} }
    = {\lim\limits_{x \to \infty } {\left( {1 + \frac{1}{x}} \right)^x} \cdot}\kern0pt {\lim\limits_{x \to \infty } {\left( {1 + \frac{1}{x}} \right)^x} \cdot}\kern0pt {\lim\limits_{x \to \infty } {\left( {1 + \frac{1}{x}} \right)^x} }
    = {e \cdot e \cdot e = {e^3}.}
    \]

    Example 3.

    Calculate the limit \(\lim\limits_{x \to \infty } {\left( {1 + {\large\frac{6}{x}}\normalsize} \right)^x}.\)

    Solution.

    Substituting \({\large\frac{6}{x}\normalsize} = {\large\frac{1}{y}\normalsize},\) so that \(x = 6y\) and \(y \to \infty\) as \(x \to \infty,\) we obtain

    \[
    {\lim\limits_{x \to \infty } {\left( {1 + \frac{6}{x}} \right)^x} }
    = {\lim\limits_{y \to \infty } {\left( {1 + \frac{1}{y}} \right)^{6y}} }
    = {\lim\limits_{y \to \infty } {\left[ {{{\left( {1 + \frac{1}{y}} \right)}^y}} \right]^6} }
    = {{\left[ {\lim\limits_{y \to \infty } {{\left( {1 + \frac{1}{y}} \right)}^y}} \right]^6} = {e^6}.}
    \]

    Page 1
    Problems 1-3
    Page 2
    Problems 4-9