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Calculus

Limits and Continuity of Functions

The Number e

Page 1
Problems 1-3
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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