Calculus

Limits and Continuity of Functions

The Number e

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

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 on problem description to see 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