Proof of the Quotient Limit Law

Why Quotients Need a Nonzero Denominator Limit

In “Limit Laws for Quotients,” the quotient limit law was stated: if \(a_n\to L\), \(b_n\to M\neq0\), and \(b_n\neq0\) for every \(n\), then \(a_n/b_n\to L/M\). The product limit law, proved in the previous tutorial, handles products, but division introduces an extra issue: the denominators must not approach zero. To prove the law, we first show that a sequence converging to a nonzero number is eventually bounded away from zero.

The condition \(b_n\neq0\) for every \(n\) ensures that the quotient sequence is defined at every index. The limit condition \(M\neq0\) has a different role: it ensures that sufficiently late denominators have absolute value at least a fixed positive number. This lower bound lets us estimate errors involving \(1/b_n\).

A Lower Bound for the Denominators

Proof. Since \(|M|/2>0\) and \(b_n\to M\), the epsilon–N definition gives an \(N_0\) such that

whenever \(n\geq N_0\). By the Reverse Triangle Inequality, established earlier in the course,

Thus the asserted lower bound holds for every \(n\geq N_0\). In particular, none of these \(b_n\) is zero. \(\square\)

This lemma is the key additional ingredient in a quotient proof. Convergence alone gives closeness to \(M\); the assumption \(M\neq0\) turns that closeness into a positive lower bound on the size of \(b_n\). The bound need only hold eventually, because convergence is determined by the terms from some index onward.

An Estimate for the Reciprocal Error

Proof. Choose \(N_0\) as in the Eventual Separation from Zero Lemma. For \(n\geq N_0\), the denominators \(b_n\) and \(M\) are both nonzero, so

The strict inequality implies the displayed weak inequality as well. This proves the estimate. \(\square\)

The estimate quantifies the effect of a small change in the denominator. Its constant depends on \(M\): if \(M\) is close to zero, the constant \(2/|M|^2\) can be large. That is not a defect in the proof. It reflects the fact that reciprocals can change substantially when their inputs are near zero.

Decomposing the Quotient Error

To compare \(a_n/b_n\) with \(L/M\), subtract the proposed limit and combine the fractions. Since \(b_n\neq0\) and \(M\neq0\), the calculation is valid:

To verify the second equality, expand its numerator: \(M(a_n-L)-L(b_n-M)=Ma_n-ML-Lb_n+LM=Ma_n-Lb_n\). The error is now separated into a term involving \(a_n-L\) and a term involving \(b_n-M\).

Proof. Apply the Triangle Inequality to the quotient-error identity. For \(n\geq N_0\), use \(|b_n|>|M|/2\):

The last inequality follows because \(1/|b_n|<2/|M|\) for \(n\geq N_0\). Replacing these strict bounds by weak bounds preserves the claimed inequality. \(\square\)

Proof of the Quotient Limit Law

Proof. Let \(\varepsilon>0\). By the Eventual Separation from Zero Lemma, choose \(N_0\) so that \(|b_n|>|M|/2\) whenever \(n\geq N_0\). Define

Since \(M\neq0\), \(C>0\), and both coefficients in the Quotient Error Estimate are at most \(C\). Because \(a_n\to L\), there is an \(N_1\) such that

whenever \(n\geq N_1\). Because \(b_n\to M\), there is an \(N_2\) such that

whenever \(n\geq N_2\). Let \(N=\max\{N_0,N_1,N_2\}\). For every \(n\geq N\), the Quotient Error Estimate gives

Thus, for every \(\varepsilon>0\), there is an \(N\) such that \(\left|a_n/b_n-L/M\right|<\varepsilon\) whenever \(n\geq N\). By the epsilon–N definition of convergence, \(a_n/b_n\to L/M\). \(\square\)

This proof allows \(L=0\). In that case, the coefficient \(2|L|/|M|^2\) is zero, so the denominator-error term in the estimate vanishes; convergence of the numerator to zero and the eventual lower bound on \(|b_n|\) are enough. The assumption \(M\neq0\), however, is essential.

Worked Examples

What to Check in a Quotient-Limit Proof

A common error is to apply the quotient limit law as soon as both numerator and denominator have limits. The denominator limit must be nonzero. Another is to confuse the condition \(b_n\neq0\) for every \(n\) with the condition \(M\neq0\): the first makes each quotient defined, while the second provides the eventual lower bound needed to control its error. In other settings, a finite number of undefined initial terms can sometimes be handled by changing those terms, but the theorem proved here assumes the quotient is defined at every index.

The proof follows a reusable sequence of steps: establish separation from zero, combine the fractions in the error, and use convergence to control the two resulting differences. The reciprocal limit theorem stated in “Limit Laws for Quotients” is another way to organize the argument: prove \(1/b_n\to1/M\), then apply the Limit of a Product. The direct estimate above makes explicit how the nonzero limit controls the denominator and proves the quotient law in one epsilon–N argument.