Kuzee
Sequences

Operations on Limits

The rules for operations on limits allow us to compute the limits of complex expressions from known limits.

The Four Arithmetic Operation Rules

Let lim⁡n→∞an=A\lim_{n \to \infty} a_n = A and lim⁡n→∞bn=B\lim_{n \to \infty} b_n = B. Then:

1. The Sum and Difference Rule

The sum and difference rule
lim⁡n→∞(an±bn)=A±B\lim_{n \to \infty} (a_n \pm b_n) = A \pm B

2. The Constant Multiple Rule

The constant multiple rule
lim⁡n→∞(c⋅an)=c⋅A(c is a constant)\lim_{n \to \infty} (c \cdot a_n) = c \cdot A \quad (c \text{ is a constant})

3. The Product Rule

The product rule
lim⁡n→∞(an⋅bn)=A⋅B\lim_{n \to \infty} (a_n \cdot b_n) = A \cdot B

4. The Quotient Rule

The quotient rule
lim⁡n→∞anbn=AB(B≠0)\lim_{n \to \infty} \frac{a_n}{b_n} = \frac{A}{B} \quad (B \neq 0)

Worked Examples

Example 1: A Fraction Limit

Find lim⁡n→∞3n2+2n+12n2−n+5\lim_{n \to \infty} \frac{3n^2 + 2n + 1}{2n^2 - n + 5}

Solution:

Divide both the numerator and the denominator by n2n^2:

lim⁡n→∞3+2n+1n22−1n+5n2=3+0+02−0+0=32\lim_{n \to \infty} \frac{3 + \frac{2}{n} + \frac{1}{n^2}}{2 - \frac{1}{n} + \frac{5}{n^2}} = \frac{3 + 0 + 0}{2 - 0 + 0} = \frac{3}{2}

Example 2: Combined Operations

Find lim⁡n→∞(2n+1n−n+3n)\lim_{n \to \infty} \left(\frac{2n+1}{n} - \frac{n+3}{n}\right)

Solution:

lim⁡n→∞(2n+1n−n+3n)=lim⁡n→∞n−2n=lim⁡n→∞(1−2n)=1\lim_{n \to \infty} \left(\frac{2n+1}{n} - \frac{n+3}{n}\right) = \lim_{n \to \infty} \frac{n-2}{n} = \lim_{n \to \infty} \left(1 - \frac{2}{n}\right) = 1

Example 3: A Product Form

Find lim⁡n→∞n+1n⋅2nn+2\lim_{n \to \infty} \frac{n+1}{n} \cdot \frac{2n}{n+2}

Solution:

lim⁡n→∞n+1n⋅2nn+2=lim⁡n→∞(n+1)⋅2nn(n+2)=lim⁡n→∞2n2+2nn2+2n=2\lim_{n \to \infty} \frac{n+1}{n} \cdot \frac{2n}{n+2} = \lim_{n \to \infty} \frac{(n+1) \cdot 2n}{n(n+2)} = \lim_{n \to \infty} \frac{2n^2 + 2n}{n^2 + 2n} = 2

Common Techniques

Technique 1: Extracting the Highest Power Term

For a fraction, divide both the numerator and the denominator by the highest power term:

\frac{a_k}{b_m}, & k = m \\ 0, & k < m \\ \infty, & k > m \end{cases}$$ ### Technique 2: Rationalization $$\lim_{n \to \infty} (\sqrt{n+1} - \sqrt{n}) = \lim_{n \to \infty} \frac{1}{\sqrt{n+1} + \sqrt{n}} = 0$$ ## Practice Problems ### Exercise 1 Find $\lim_{n \to \infty} \frac{5n^3 - 2n + 1}{3n^3 + n^2}$ <ReferenceAnswer title="Reference Answer" tags={['operations on sequence limits']}> **Solution**: Divide both the numerator and the denominator by $n^3$: $$\lim_{n \to \infty} \frac{5 - \frac{2}{n^2} + \frac{1}{n^3}}{3 + \frac{1}{n}} = \frac{5}{3}$$ **Answer**: $\frac{5}{3}$ </ReferenceAnswer> ### Exercise 2 Find $\lim_{n \to \infty} \frac{n^2 + 1}{n + 1}$ <ReferenceAnswer title="Reference Answer" tags={['operations on sequence limits']}> **Solution**: The degree of the numerator is higher than that of the denominator, so the limit is infinity: $$\lim_{n \to \infty} \frac{n^2 + 1}{n + 1} = \lim_{n \to \infty} \frac{n(n + \frac{1}{n})}{n(1 + \frac{1}{n})} = \lim_{n \to \infty} \frac{n + \frac{1}{n}}{1 + \frac{1}{n}} = \infty$$ **Answer**: divergent (tends to infinity) </ReferenceAnswer> ### Exercise 3 Find $\lim_{n \to \infty} (\sqrt{n^2 + n} - n)$ <ReferenceAnswer title="Reference Answer" tags={['operations on sequence limits']}> **Solution**: Rationalize: $$\lim_{n \to \infty} (\sqrt{n^2 + n} - n) = \lim_{n \to \infty} \frac{(\sqrt{n^2 + n} - n)(\sqrt{n^2 + n} + n)}{\sqrt{n^2 + n} + n}$$ $$= \lim_{n \to \infty} \frac{n^2 + n - n^2}{\sqrt{n^2 + n} + n} = \lim_{n \to \infty} \frac{n}{\sqrt{n^2 + n} + n}$$ $$= \lim_{n \to \infty} \frac{1}{\sqrt{1 + \frac{1}{n}} + 1} = \frac{1}{2}$$ **Answer**: $\frac{1}{2}$ </ReferenceAnswer> --- ## Summary ### Symbols Used in This Article | 符号 | 类型 | 读音/说明 | 在本文中的含义 | | :--- | :--- | :--- | :--- | | $a_n, b_n$ | 数学符号 | a/b-sub-n | Sequences | | $A, B$ | 数学符号 | A, B | The limiting values of the sequences | | $c$ | 数学符号 | c | A constant | | $\lim_{n \to \infty}$ | 数学符号 | limit as n approaches infinity | The limit as $n$ tends to infinity | | $\pm$ | 数学符号 | plus or minus | Plus or minus | | $\sqrt{n}$ | 数学符号 | square root of n | The square root of $n$ | | $\infty$ | 数学符号 | infinity | Infinity | | $n^k, n^m$ | 数学符号 | n to the k/m | Powers of $n$ | ### 中英对照 | 中文术语 | 英文术语 | 音标 | 说明 | | :--- | :--- | :--- | :--- | | 四则运算 | four arithmetic operations | /fɔː ˌærɪθˈmetɪk ˌɒpəˈreɪʃənz/ | Addition, subtraction, multiplication and division | | 有理化 | rationalization | /ˌræʃənəlaɪˈzeɪʃən/ | The technique of eliminating radicals | | 收敛数列 | convergent sequence | /kənˈvɜːdʒənt ˈsiːkwəns/ | A sequence with a limit | | 商法则 | quotient rule | /ˈkwəʊʃənt ruːl/ | The division operation of limits |
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Properties of Limits
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Convergence Criteria