Calculus II · Unit 4A · lesson

Limit Laws for Sequences

Concept

Learning objectives

use algebraic limit laws to combine convergent sequences and identify indeterminate forms that require more work.

Limit Laws for Sequences

Explanation

Algebra works after convergence is established

If two sequences converge, their sums, differences, products, and appropriate quotients behave as expected. These laws let us replace complicated expressions by limits of simpler pieces. The laws do not say that every expression with infinity symbols can be manipulated like arithmetic. "Infinity over infinity" is not a number; it is a warning that dominant growth must be compared.

A reliable strategy is to normalize by the largest power, factor a dominant term, rationalize a radical, or use a known function limit. Before applying a quotient law, verify that the denominator limit is nonzero. Before moving a continuous function through a limit, check that the function is continuous at the limiting value and that the sequence stays in its domain.

Bridge

Why algebra survives the limiting process

If two sequences settle near $A$ and $B$ , then their late sums, products, and valid quotients should settle near the corresponding algebraic combinations. The limit laws formalize this stability and reduce complicated sequences to simpler components.

The laws are not permission to substitute blindly. Division requires a nonzero limiting denominator, and roots require valid domains. Identify component limits, verify the operation is defined at those limits, and only then combine them.

Proof idea

The sum law is an error estimate

If $a_n\to A$ and $b_n\to B$ , then

|(a_n+b_n)-(A+B)|\le |a_n-A|+|b_n-B|.

Making each component error less than $\varepsilon/2$ makes the total error less than $\varepsilon$ .

Concept

Limit-law summary

If $a_n\to A$ and $b_n\to B$ , then

a_n\pm b_n\to A\pm B,\quad a_nb_n\to AB,

and $a_n/b_n\to A/B$ when $B\ne0$ . Continuous functions preserve limits where defined.

Guided walkthrough

A radical sequence

Evaluate $a_n=\sqrt{n^2+4n}-n$ . Rationalize:

a_n=\frac{4n}{\sqrt{n^2+4n}+n} =\frac{4}{\sqrt{1+4/n}+1}\to2.

The original form $\infty-\infty$ concealed a finite difference.

Worked example

A quotient needs a domain check

Evaluate

\lim_{n\to\infty}\frac{2n^2-5n}{n^2+3}.

Divide by $n^2$ :

\frac{2-5/n}{1+3/n^2}.

The numerator tends to $2$ and the denominator to $1$ . Since the limiting denominator is nonzero, the quotient law applies and the limit is $2$ .

Common mistake

The quotient law does not evaluate zero over zero

When numerator and denominator both approach zero, the quotient law is inconclusive. Rewrite or compare rates instead.

Interactive checku4a-sequence_limit_laws-01

Evaluate $\lim_{n\to\infty}\frac{3n^2-1}{n^2+5}$ .

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Show hint

Divide by $n^2$ .

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Exercise

Evaluate $(n^2+1)^{1/2}/n$ .

Exercise

Find $\lim (2+1/n)^3$ .

Exercise

Explain why the quotient law cannot be applied to $n/(1/n)$ using a denominator limit of zero.

Exercise

Rationalize $\sqrt{n^2+n}-n$ .

After the explanation

Use the section idea

Reading lens

Track the integer domain, late-term behavior, monotonicity, bounds, and any recurrence before asserting a limit.

Mental model

A sequence converges when every sufficiently late term remains arbitrarily close to one finite target.

Decision

Use algebraic limits when a formula is explicit; use bounds and monotonicity when a recurrence hides the formula.

Common trap

Reading a finite plot as proof or solving a recurrence's fixed-point equation before proving convergence.

Check yourself

Can you justify both the candidate limit and why the terms must approach it?

Source & rights

Original instruction with traceable references.

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