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

Direct Substitution

Direct substitution is the most basic and direct method for evaluating limits. When a function is continuous at the limit point, we can directly substitute the value of the limit point into the function to evaluate it.

Basic Principle

Basic principle of direct substitution

If the function $f(x)$ is continuous at the point $a$ , then:

$\lim_{x \to a} f(x) = f(a)$

Applicable Conditions

1. The Function Is Continuous at the Limit Point

The function is defined at the point
The limit of the function at the point equals its value
The left-hand limit equals the right-hand limit at the point

2. Common Continuous Functions

Polynomial functions: $P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$
Rational functions: $R(x) = \frac{P(x)}{Q(x)}$ (where $Q(a) \neq 0$ )
Exponential functions: $e^x$
Logarithmic functions: $\ln x$ ( $x > 0$ )
Trigonometric functions: $\sin x, \cos x, \tan x$ , etc.

Solving Steps

Check whether the function is defined at the limit point
Determine whether the function is continuous at the point
Substitute directly to evaluate

Worked Examples

Example 1

Find the limit $\lim_{x \to 3} (2x^2 - 5x + 1)$

Reference Answer(1 个标签)

direct substitution

Idea: This is a polynomial function, which is continuous everywhere, so we can substitute directly.

Detailed steps:

Check the type of function: $f(x) = 2x^2 - 5x + 1$ is a polynomial function
A polynomial function is continuous everywhere
Substitute $x = 3$ directly: $f(3) = 2(3)^2 - 5(3) + 1 = 2(9) - 15 + 1 = 18 - 15 + 1 = 4$

Answer: $\lim_{x \to 3} (2x^2 - 5x + 1) = 4$

Example 2

Find the limit $\lim_{x \to 2} \frac{x^2 + 3x - 2}{x - 1}$

Reference Answer(1 个标签)

direct substitution

Idea: This is a rational function, so we need to check whether the denominator is zero.

Detailed steps:

Check the denominator: when $x = 2$ , the denominator $x - 1 = 2 - 1 = 1 \neq 0$
A rational function is continuous wherever the denominator is nonzero
Substitute $x = 2$ directly: $\frac{2^2 + 3(2) - 2}{2 - 1} = \frac{4 + 6 - 2}{1} = \frac{8}{1} = 8$

Answer: $\lim_{x \to 2} \frac{x^2 + 3x - 2}{x - 1} = 8$

Notes

1. When the Denominator Is Zero

If substitution makes the denominator zero, direct substitution cannot be used; other methods are needed.

2. When the Radicand Is Negative

If substitution makes the radicand negative, the function is undefined at the point.

3. Logarithmic Functions

The argument of a logarithmic function must be positive.

Practice Problems

Exercise 1

Find the limit $\lim_{x \to 1} (x^3 - 2x^2 + x + 5)$

Reference Answer(1 个标签)

direct substitution

Idea: A polynomial function, so substitute directly.

Detailed steps:

$f(x) = x^3 - 2x^2 + x + 5$ is a polynomial function
A polynomial function is continuous everywhere
Substitute $x = 1$ directly: $f(1) = 1^3 - 2(1)^2 + 1 + 5 = 1 - 2 + 1 + 5 = 5$

Answer: $\lim_{x \to 1} (x^3 - 2x^2 + x + 5) = 5$

Exercise 2

Find the limit $\lim_{x \to 0} \frac{\sin x}{x}$

Reference Answer(1 个标签)

direct substitution

Idea: This is an important limit; it cannot be evaluated by direct substitution, so we need other methods.

Detailed steps:

When $x = 0$ , the numerator $\sin 0 = 0$ and the denominator $x = 0$
This is a $\frac{0}{0}$ indeterminate form, so direct substitution cannot be used
We need to use the important limit: $\lim_{x \to 0} \frac{\sin x}{x} = 1$

Answer: $\lim_{x \to 0} \frac{\sin x}{x} = 1$

Summary

Symbols Used in This Article

符号	类型	读音/说明	在本文中的含义
$\lim_{x \to a} f(x)$	数学符号	limit as x approaches a	The limit of the function as $x \to a$
$f(a)$	数学符号	f of a	The value of the function at the point $a$
$a$	数学符号	a	The limit point
$P(x)$	数学符号	P of x	A polynomial function
$Q(x)$	数学符号	Q of x	The denominator of a rational function
$e^x$	数学符号	e to the x	The exponential function
$\ln x$	数学符号	natural logarithm of x	The natural logarithmic function
$\sin x, \cos x, \tan x$	数学符号	sine/cosine/tangent of x	Trigonometric functions
$\frac{0}{0}$	数学符号	zero over zero	Indeterminate form

中英对照

中文术语	英文术语	音标	说明
直接代入法	direct substitution	/dəˈrekt ˌsʌbstɪˈtjuːʃən/	The method of directly substituting the limit point
连续函数	continuous function	/kənˈtɪnjuəs ˈfʌŋkʃən/	A function with no breaks
多项式函数	polynomial function	/ˌpɒlɪˈnəʊmiəl ˈfʌŋkʃən/	A function that is a polynomial
有理函数	rational function	/ˈræʃənəl ˈfʌŋkʃən/	A function that is a polynomial divided by a polynomial
指数函数	exponential function	/ˌekspəˈnenʃəl ˈfʌŋkʃən/	A function of the form $a^x$
对数函数	logarithmic function	/ˌlɒɡəˈrɪðmɪk ˈfʌŋkʃən/	A function involving logarithms
三角函数	trigonometric function	/ˌtrɪɡənəˈmetrɪk ˈfʌŋkʃən/	Functions such as sine and cosine
不定式	indeterminate form	/ˌɪndɪˈtɜːmɪnət fɔːm/	A limit form that cannot be determined directly

Methods for Evaluating Limits

Factorization Method