This is a beta course, so its structure, chapters, and examples may continue to change.

The Second Important Limit

The second important limit is another fundamental tool in limit computation. It establishes the relationship between the natural constant $e$ and limit operations, with important applications in finance, probability theory, and other fields.

The Limit Expression

The second important limit

The expression of the second important limit is:

$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$

where $e \approx 2.71828$ is the base of the natural logarithm.

Geometric meaning: as $x$ tends to infinity, $\left(1 + \frac{1}{x}\right)^x$ tends to the natural constant $e$ . This limit has important applications in compound interest, continuously compounded interest, and other financial problems.

Geometric Meaning

The geometric meaning of this limit is: as $x$ tends to infinity, $\left(1 + \frac{1}{x}\right)^x$ tends to the natural constant $e$ . This limit has important applications in compound interest, continuously compounded interest, and other financial problems.

Interpretation through Compound Interest

In compound interest calculations:

If the annual interest rate is $r$ and interest is compounded $n$ times per year
Then the amount after $t$ years is: $A = P\left(1 + \frac{r}{n}\right)^{nt}$
As $n \to \infty$ , we obtain continuously compounded interest: $A = Pe^{rt}$

Ideas for the Proof

1. Sequence Form

First prove that the sequence $x_n = \left(1 + \frac{1}{n}\right)^n$ converges to $e$

2. Monotone Convergence Criterion

One can prove that the sequence is monotonically increasing
One can prove that the sequence is bounded above
Therefore the sequence converges

3. Function Form

Use the sequence limit to extend to the function limit

Generalized Forms

Generalized forms of the second important limit

1. General Form

$\lim_{x \to \infty} \left(1 + \frac{a}{x}\right)^x = e^a$

Proof: $\lim_{x \to \infty} \left(1 + \frac{a}{x}\right)^x = \lim_{x \to \infty} \left(1 + \frac{1}{\frac{x}{a}}\right)^{\frac{x}{a} \cdot a} = e^a$

2. Composite Form

$\lim_{x \to \infty} \left(1 + \frac{1}{f(x)}\right)^{f(x)} = e$

where $\lim_{x \to \infty} f(x) = \infty$

3. Reciprocal Form

$\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e$

Proof:

Let $t = \frac{1}{x}$ ; then as $x \to 0$ , $t \to \infty$
$\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = \lim_{t \to \infty} \left(1 + \frac{1}{t}\right)^t = e$

Worked Examples

Example 1

Find $\lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^x$

Solution: $\lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^x = e^2$

Example 2

Find $\lim_{x \to 0} (1 + x)^{\frac{1}{x}}$

Solution:

Let $t = \frac{1}{x}$ ; then as $x \to 0$ , $t \to \infty$
$\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = \lim_{t \to \infty} \left(1 + \frac{1}{t}\right)^t = e$

Example 3

Find $\lim_{x \to \infty} \left(1 + \frac{1}{x^2}\right)^{x^2}$

Solution:

Let $t = x^2$ ; then as $x \to \infty$ , $t \to \infty$
$\lim_{x \to \infty} \left(1 + \frac{1}{x^2}\right)^{x^2} = \lim_{t \to \infty} \left(1 + \frac{1}{t}\right)^t = e$

Financial Applications

Continuously Compounded Interest

In finance, the formula for continuously compounded interest is: $A = P \cdot e^{rt}$

where:

$A$ is the final amount
$P$ is the principal
$r$ is the annual interest rate
$t$ is the time (in years)

Present Value

The present value formula is: $P = A \cdot e^{-rt}$

Memory Tips

Mnemonics

One plus the reciprocal raised to a power, the limit equals e
Key: remember the form $\left(1 + \frac{1}{x}\right)^x$

Generalization Memory

The generalization of the second important limit: $\left(1 + \frac{a}{x}\right)^x = e^a$

Summary

Greek Letters Used in This Article

This article uses several Greek letters to denote mathematical concepts. Here is a summary:

希腊字母	符号	英文名	中文读音	在文中的含义
Epsilon	$\varepsilon$	Epsilon	伊普西隆	An arbitrarily small positive number
Nu	$n$	Nu	缪	A positive integer

Practice Problems

Exercise 1

Find the limit $\lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^x$ .

Reference Answer(1 个标签)

important limit

Idea: Use the generalized form of the second important limit.

Detailed steps:

$\lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^x = e^3$

Answer: The limit is $e^3$ .

Exercise 2

Find the limit $\lim_{x \to 0} (1 + 2x)^{\frac{1}{x}}$ .

Reference Answer(1 个标签)

important limit

Idea: Use a change of variable and the second important limit.

Detailed steps:

Let $t = \frac{1}{x}$ ; then as $x \to 0$ , $t \to \infty$
$\lim_{x \to 0} (1 + 2x)^{\frac{1}{x}} = \lim_{t \to \infty} \left(1 + \frac{2}{t}\right)^t = e^2$

Answer: The limit is $e^2$ .

Exercise 3

Find the limit $\lim_{x \to \infty} \left(1 + \frac{1}{x+1}\right)^x$ .

Reference Answer(1 个标签)

important limit

Idea: Use a change of variable and the second important limit.

Detailed steps:

Let $t = x + 1$ ; then as $x \to \infty$ , $t \to \infty$
$\lim_{x \to \infty} \left(1 + \frac{1}{x+1}\right)^x = \lim_{t \to \infty} \left(1 + \frac{1}{t}\right)^{t-1}$
$= \lim_{t \to \infty} \left(1 + \frac{1}{t}\right)^t \cdot \left(1 + \frac{1}{t}\right)^{-1} = e \cdot 1 = e$

Answer: The limit is $e$ .

Summary

Symbols Used in This Article

符号	类型	读音/说明	在本文中的含义
$e$	数学符号	natural constant	The base of the natural logarithm, approximately 2.71828
$\lim$	数学符号	limit	The limit of a function or sequence
$\to$	数学符号	tends to	A variable tending to some value
$\infty$	数学符号	infinity	Infinity

中英对照

中文术语	英文术语	音标	说明
第二个重要极限	second important limit	/ˈsekənd ɪmˈpɔːtənt ˈlɪmɪt/	Another fundamental tool in limit computation
自然常数	natural constant	/ˈnætʃərəl ˈkɒnstənt/	The base of the natural logarithm $e$
自然对数	natural logarithm	/ˈnætʃərəl ˈlɒɡərɪðəm/	The logarithm with base $e$
复利	compound interest	/ˈkɒmpaʊnd ˈɪntrəst/	A way of computing interest
连续复利	continuous compound interest	/kənˈtɪnjuəs ˈkɒmpaʊnd ˈɪntrəst/	The limiting case of compound interest
单调递增	monotone increasing	/ˈmɒnətəʊn ɪnˈkriːsɪŋ/	Sequence or function values gradually increase
有上界	bounded above	/ˈbaʊndɪd əˈbʌv/	There exists a number greater than or equal to all terms

The First Important Limit

Limits of Exponential Functions