Limit Laws for Sequences

Why Limit Laws Matter

Sequences often arise from expressions that combine simpler sequences. If the component sequences converge, limit laws let us find the limit of the combined expression without starting from scratch each time. The rules resemble familiar algebra, but each has conditions. In particular, division requires the denominator sequence to approach a nonzero number.

We will use the epsilon-N definition: a sequence \((a_n)\) converges to \(L\) if, for every \(\varepsilon>0\), there is an \(N\in\mathbb{N}\) such that \(n\ge N\) implies \(|a_n-L|<\varepsilon\). We write \(\lim_{n\to\infty}a_n=L\). The definition says that terms can be made arbitrarily close to \(L\) by taking the index sufficiently large.

A basic example used below is \(1/n\to 0\). Given \(\varepsilon>0\), choose \(N\in\mathbb{N}\) with \(N>1/\varepsilon\). Then \(n\ge N\) gives \(0<1/n\le 1/N<\varepsilon\), which verifies the limit directly.

Sum, Difference, and Constant-Multiple Laws

The sum law follows by measuring how far \(a_n+b_n\) is from the proposed limit \(L+M\). The difference law uses the same idea, and the constant-multiple law scales the distance from the limit. Here are the rules together:

For the sum, let \(\varepsilon>0\). By convergence, there are indices after which \(|a_n-L|<\varepsilon/2\) and \(|b_n-M|<\varepsilon/2\). Once both conditions hold, the triangle inequality gives

This proves the sum law from the definition. For the difference, observe that \(|(a_n-b_n)-(L-M)|\le |a_n-L|+|b_n-M|\), so the same choice of indices proves the result. For a constant multiple, if \(c=0\), the sequence \(ca_n\) is constantly zero and has limit \(0=cL\). If \(c\ne 0\), choose indices for which \(|a_n-L|<\varepsilon/|c|\). Then \(|ca_n-cL|=|c||a_n-L|<\varepsilon\). Thus the constant-multiple law also follows directly from the definition.

The Product Law

The product law is \(\lim(a_nb_n)=LM\). Unlike the sum proof, it needs one extra observation: a convergent sequence is eventually bounded. Indeed, because \(a_n\to L\), the definition with \(\varepsilon=1\) gives an index after which \(|a_n-L|<1\). The triangle inequality then gives \(|a_n|\le |L|+1\) for all such indices.

To prove the product law, rewrite the difference from the proposed limit as

Let \(\varepsilon>0\). For all sufficiently large \(n\), we have \(|a_n|\le |L|+1\). Convergence also lets us require both \(|b_n-M|<\varepsilon/[2(|L|+1)]\) and \(|a_n-L|<\varepsilon/[2(|M|+1)]\). Taking \(n\) large enough to meet all three conditions,

This is the epsilon-N condition for \(a_nb_n\to LM\), proving the product law. The boundedness step matters: convergence of \(b_n\) controls \(b_n-M\), but the product also contains the factor \(a_n\), which must not grow without bound.

The Quotient Law and Its Condition

Division requires special care. If \(b_n\to M\ne0\), then \(b_n\) is eventually bounded away from zero. Specifically, convergence with tolerance \(|M|/2\) gives, for all sufficiently large \(n\),

So the denominator is nonzero from some index onward. To prove the quotient limit, use the identity

For a given \(\varepsilon>0\), take \(n\) large enough that \(|b_n|>|M|/2\) and that

These conditions are available because both sequences converge. Substituting the bounds yields

This proves the quotient law. The condition \(M\ne0\) is essential to the argument: it provides a positive lower bound for \(|b_n|\). If the quotient is undefined at some initial indices, its tail is still defined once \(b_n\ne0\); the limit concerns the terms from that point onward.

A Common Mistake: Dividing Limits When the Denominator Limit Is Zero

A tempting but invalid calculation is to apply the quotient law whenever the numerator and denominator have limits, without checking the denominator limit. For example, take \(a_n=1/n\) and \(b_n=1/n\). Both sequences tend to zero. The incorrect step would be to claim

The quotient law does not apply because the denominator limit is zero, and \(0/0\) is not a defined number. In this example, direct simplification gives \(a_n/b_n=(1/n)/(1/n)=1\) for every \(n\), so the quotient limit is \(1\). Other quotients with numerator and denominator both tending to zero can behave differently. The correct procedure is to check that the denominator limit is nonzero before using the quotient law; otherwise, analyze the quotient itself.

Using the Laws Carefully

The limit laws allow algebraic expressions to be handled in stages. First identify the limit of each component sequence. Then use the appropriate law, verifying any special condition—most importantly, that a denominator limit is nonzero. The laws give a limit only when their hypotheses are met; they do not turn an indeterminate expression into a valid computation.

These rules are useful throughout the study of sequences because they turn many complicated limit questions into simpler ones. Their epsilon-N justification also explains why they work: when each component is close to its limit, the combined expression is close to the corresponding combination of those limits, provided the operations remain valid.