Calculus II · Unit 4A · lesson
Limit Laws for Sequences
Apply sequence limit laws safely, check denominators and domains, and understand how algebraic operations preserve eventual behavior.
Section overview
Sequences and their limitsWhat this section is building
Apply sequence limit laws safely, check denominators and domains, and understand how algebraic operations preserve eventual behavior.
A sequence converges when every sufficiently late term remains arbitrarily close to one finite target.
Use algebraic limits when a formula is explicit; use bounds and monotonicity when a recurrence hides the formula.
Reading a finite plot as proof or solving a recurrence's fixed-point equation before proving convergence.
Learning objectives
use algebraic limit laws to combine convergent sequences and identify indeterminate forms that require more work.
Limit Laws for Sequences
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.
Why algebra survives the limiting process
If two sequences settle near and , 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.
The sum law is an error estimate
If and , then
Making each component error less than makes the total error less than .
Limit-law summary
If and , then
and when . Continuous functions preserve limits where defined.
A radical sequence
Evaluate . Rationalize:
The original form concealed a finite difference.
A quotient needs a domain check
Evaluate
Divide by :
The numerator tends to and the denominator to . Since the limiting denominator is nonzero, the quotient law applies and the limit is .
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.
u4a-sequence_limit_laws-01Evaluate .
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Divide by .
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Evaluate .
Find .
Explain why the quotient law cannot be applied to using a denominator limit of zero.
Rationalize .
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