Limit Comparison Arguments

From a Quotient Limit to a Comparison

In the previous tutorial, fixed-multiple comparisons were used to transfer boundedness between sequences. A useful way to obtain such comparisons is to study the limit of their quotient. If two sequences are eventually positive and their quotient tends to a positive finite number, then their terms are eventually within fixed positive multiples of one another. The quotient need not equal a constant at any particular index; being sufficiently close to a positive constant is enough.

The positivity and finiteness of the quotient limit matter. They provide both an eventual upper bound and an eventual positive lower bound. These bounds can transfer more information than a one-sided comparison alone.

A quotient defined only for sufficiently large indices is enough: convergence depends on the tail, and finite changes preserve convergence. In the comparison arguments below, we will assume the sequences are eventually positive so that multiplying or dividing inequalities does not reverse their direction.

The Two-Sided Estimate

Suppose \(a_n/b_n\to c\), with \(c>0\). Apply the definition of convergence using the tolerance \(c/2\). Eventually the quotient differs from \(c\) by less than \(c/2\), so it lies between \(c/2\) and \(3c/2\). Multiplying by the positive term \(b_n\) gives a two-sided estimate for \(a_n\).

Proof. Since \(a_n/b_n\to c\), there is an index \(N_1\) such that, for every \(n\geq N_1\),

This inequality implies

for every \(n\geq N_1\). Choose \(N\) large enough that this estimate holds and both sequences are positive for \(n\geq N\). Multiplication by \(b_n>0\) gives the asserted two-sided bounds (with weak inequalities as well).

For boundedness, the upper bound \(a_n\leq(3c/2)b_n\) on the tail is a fixed-multiple magnitude comparison. If \((b_n)\) is bounded, the Boundedness Transfers Through an Upper Magnitude Comparison theorem from the previous tutorial shows that \((a_n)\) is bounded. The lower bound can be rearranged, since \(c/2>0\), to give \(b_n\leq(2/c)a_n\) on the tail. The same theorem shows that boundedness of \((a_n)\) implies boundedness of \((b_n)\). Thus the sequences are bounded together or unbounded together.

Now suppose \(b_n\to+\infty\). Given any real \(R\), choose a positive number \(T\) such that \((c/2)T>R\). For all sufficiently large \(n\), both \(n\geq N\) and \(b_n>T\), and hence

Therefore \(a_n\to+\infty\). Conversely, the upper bound gives \(b_n\geq(2/(3c))a_n\) on the tail. If \(a_n\to+\infty\), this inequality proves \(b_n\to+\infty\) by the same threshold argument, using \(2/(3c)>0\). This proves both directions. \(\square\)

The result is a limit-based method for obtaining the fixed constants required by comparison arguments. It does not say that the sequences have the same limit: a quotient tending to \(c\) says their sizes are related by a factor approaching \(c\), and their finite limits, when they exist, may therefore differ by that factor.

Using the Ratio to Find a Finite Limit

A positive finite ratio limit is also useful when one of the sequences converges to a finite value. On a tail, the identity \(a_n=(a_n/b_n)b_n\) expresses one sequence as the product of the quotient and the other sequence. The Limit of a Product theorem then determines the limit of \(a_n\).

Proof. Choose an index \(N\) such that \(b_n>0\) for every \(n\geq N\), and define \(r_n=a_n/b_n\) on that tail. Extend \((r_n)\) to a real sequence by assigning arbitrary real values to its finitely many terms before \(N\). The extension still converges to \(c\), since finite changes preserve convergence. For every \(n\geq N\),

By the Limit of a Product theorem, \(r_nb_n\to cL\). The sequence \((a_n)\) agrees with \((r_nb_n)\) for every \(n\geq N\), so the Finite Changes Preserve Convergence theorem implies \(a_n\to cL\). \(\square\)

The conclusion includes the case \(L=0\): then \(a_n\to0\), even if the ratio limit \(c\) is not \(1\). When \(c=1\), the theorem says that \(a_n\) and \(b_n\) have the same finite limit, provided the quotient is defined on a positive tail.

Why the Positive Finite Condition Matters

A quotient limit of zero does not give a positive lower comparison. It may still provide useful information in other settings, but it cannot be used here to conclude that the sequences share boundedness status or divergence to infinity. Likewise, a quotient that grows without bound does not produce the fixed upper and lower constants in the theorem. Always check which kind of quotient limit has actually been established before using a limit comparison.

There are two further common pitfalls. First, the quotient must be defined on a tail; if \(b_n=0\) at arbitrarily large indices, the ratio \(a_n/b_n\) cannot have the stated ordinary real limit along all sufficiently large indices. Second, the comparison step uses \(b_n>0\). Without eventual positivity, multiplying a quotient inequality by \(b_n\) can reverse its direction, and quotient estimates alone do not give the displayed order bounds. For sequences that change sign, one must make an appropriate comparison of magnitudes and verify the needed hypotheses.

A Practical Limit-Comparison Routine

When a quotient appears simpler than the sequences themselves, use it to generate the constants needed for a comparison. Then match the comparison to the desired conclusion rather than treating all limit behavior as interchangeable.

A ratio limit is a way to turn information about relative size into concrete inequalities. A positive finite limit gives comparisons in both directions; a zero limit loses the lower bound, and a limit that is not finite loses the fixed upper bound. Keeping these distinctions explicit makes limit comparisons reliable and prevents a useful estimate from being applied beyond what it proves.