Why Leading Powers Matter
When a sequence is defined by a polynomial or a ratio of polynomials in its index, its terms may become large, small, or approach a finite value. The main question is which terms control that behavior as the index increases.
For large positive indices, a polynomial’s highest-power term eventually dominates its lower-power terms. For a ratio of polynomials, comparing the highest powers in the numerator and denominator usually determines the limit. The leading coefficients then determine the value of a finite limit or the direction of unbounded growth.
This method gives an efficient way to evaluate many sequence limits, but it requires checking that the expression is defined for all sufficiently large indices.
Polynomial Sequences
A nonzero polynomial \(P(n)\) can be written as \(P(n)=a_m n^m+a_{m-1}n^{m-1}+\cdots+a_1n+a_0\), where \(a_m\ne 0\). Its degree is \(m\), and its leading coefficient is \(a_m\).
To identify its behavior as \(n\to\infty\), divide each term by \(n^m\). The leading term becomes the constant \(a_m\), while every lower-degree term contains a factor \(1/n^j\) for some positive integer \(j\), which approaches zero. Thus \(P(n)/n^m\) approaches \(a_m\).
In particular, if \(m\ge 1\), the polynomial’s magnitude grows without bound. Its sign for sufficiently large \(n\) is the sign of its leading coefficient, because the lower-degree terms become small relative to the leading term.
Worked Example: A Polynomial Sequence
Evaluate \(\lim_{n\to\infty}(4n^3-7n+2)\).
The highest power is \(n^3\), with leading coefficient \(4\). Factor out \(n^3\):
The expression in parentheses approaches \(4\), so it is positive for all sufficiently large \(n\). Also, \(n^3\to+\infty\). Therefore the product grows without bound:
Check: dividing the original expression by \(n^3\) gives \(4-7/n^2+2/n^3\to4>0\). Thus the polynomial is eventually positive and has the size of a positive constant times \(n^3\), confirming the limit \(+\infty\).
Rational Sequences and Their Domains
A rational expression in \(n\) has the form \(P(n)/Q(n)\), where \(P\) and \(Q\) are polynomials and \(Q\) is not the zero polynomial. The expression is defined at an index \(n\) only when \(Q(n)\ne0\).
A nonzero polynomial has only finitely many real zeros. Therefore, if \(Q\) is not the zero polynomial, \(P(n)/Q(n)\) is defined for all sufficiently large positive integers \(n\), even if the denominator vanishes at one or more earlier indices. A sequence limit concerns the tail of the sequence, so finitely many initial terms do not change a limit. If the original sequence is required to be defined starting at \(n=1\), however, an index with zero denominator must be excluded or the sequence must be defined from a later starting index.
The phrase “grows without bound” here means the sequence approaches \(+\infty\) or \(-\infty\), rather than a finite real number. The sign conclusion uses positive indices tending to \(+\infty\).
Why Comparing Degrees Works
The degree comparison follows by factoring out the highest power of \(n\) from each polynomial. Suppose \(P\) has degree \(m\) and \(Q\) has degree \(k\). Then we can write
and
The terms involving \(1/n\) and its positive powers approach zero. Since \(b_k\ne0\), the parenthetical denominator approaches a nonzero number and is nonzero for all sufficiently large \(n\). Dividing gives
The fraction on the right approaches \(a_m/b_k\), which is nonzero. The remaining factor \(n^{m-k}\) determines whether the whole expression approaches zero, a nonzero finite value, or grows without bound.
- If \(m<k\), then \(\lim_{n\to\infty}P(n)/Q(n)=0\).
- If \(m=k\), then \(\lim_{n\to\infty}P(n)/Q(n)=a_m/b_k\).
- If \(m>k\) and \(a_m/b_k>0\), then \(P(n)/Q(n)\to+\infty\).
- If \(m>k\) and \(a_m/b_k<0\), then \(P(n)/Q(n)\to-\infty\).
These conclusions also follow directly from the factored form. When \(m<k\), \(n^{m-k}=1/n^{k-m}\to0\), and the other factor approaches a finite nonzero constant. When \(m=k\), the power of \(n\) is \(1\). When \(m>k\), \(n^{m-k}\to+\infty\), and the other factor approaches the nonzero constant \(a_m/b_k\), which determines the sign.
Worked Examples
Worked Example: Equal Degrees Give a Finite Limit
Evaluate \(\lim_{n\to\infty}(3n^3-2n+5)/(6n^3+n^2-4)\).
Both polynomials have degree \(3\). Their leading coefficients are \(3\) and \(6\), so the degree rule predicts the limit \(3/6=1/2\). To verify it by dividing each term by \(n^3\),
The numerator approaches \(3\), and the denominator approaches \(6\ne0\). By the quotient limit law, the ratio approaches \(3/6\):
Check: the denominator divided by \(n^3\) approaches \(6\), so it is nonzero for all sufficiently large \(n\); the divided numerator and denominator have the stated limits. The resulting ratio limit is \(1/2\), as predicted by the leading coefficients.
Worked Example: A Lower-Degree Numerator Gives Zero
Evaluate \(\lim_{n\to\infty}(5n^2+1)/(2n^4-3n+7)\).
The numerator has degree \(2\), while the denominator has degree \(4\). Divide both by \(n^4\):
The numerator approaches \(0+0=0\). The denominator approaches \(2\ne0\). Therefore,
Check: the denominator divided by \(n^4\) tends to \(2\), so it is nonzero eventually; the numerator divided by \(n^4\) tends to \(0\). The quotient limit law confirms the result.
Worked Example: A Higher-Degree Numerator Gives Negative Unbounded Growth
Evaluate the behavior of \(\lim_{n\to\infty}(-2n^4+3n)/(3n^2+1)\).
The numerator degree is \(4\), greater than the denominator degree \(2\). The leading-coefficient ratio is \(-2/3<0\), suggesting a limit of \(-\infty\). To make the direction explicit, divide the ratio by \(n^2\) in its numerator and denominator:
The parenthetical factor approaches \(-2/3\). Hence it is negative and bounded away from zero for all sufficiently large \(n\), while \(n^2\to+\infty\). Their product therefore approaches \(-\infty\):
Check: dividing the original expression by \(n^2\) leaves a factor that tends to \(-2/3\), so the original expression has the size of a negative constant times \(n^2\). This verifies both unboundedness and its negative direction.
Denominator Zeros and a Common Mistake
For example, \((n^2-4)/(n-2)\) is undefined at \(n=2\). For \(n\ge3\), the denominator is nonzero and factoring gives
Thus, if the sequence is taken from \(n=3\) onward, its limit is \(+\infty\). Cancellation does not make the original expression defined at \(n=2\); it only shows that the expressions agree where \(n\ne2\). A finite number of omitted indices does not change the limit of the tail.
A common mistake is to compare leading coefficients without first comparing degrees. For instance, a student might claim that
This is incorrect: the numerator has degree \(2\) and the denominator degree \(1\), so their leading-coefficient ratio alone is not the limit. Divide the expression by \(n\) after factoring its highest powers:
The fraction in parentheses approaches \(2\), while \(n\to+\infty\). The correct limit is therefore \(+\infty\), not \(2\). The ratio has the size of \(2n\), which confirms the corrected conclusion.
A Reliable Procedure
For a rational sequence, confirm that the denominator polynomial is not identically zero and that the expression is defined for all sufficiently large indices.
Identify the highest power with a nonzero coefficient in the numerator and denominator.
A smaller numerator degree gives zero; equal degrees give the ratio of leading coefficients; a larger numerator degree gives unbounded behavior.
For a higher-degree numerator, the sign of the leading-coefficient ratio determines whether the limit is \(+\infty\) or \(-\infty\).
This method matters because it turns a complicated-looking expression into a comparison of dominant terms. It is also a useful check on algebra: a proposed finite limit cannot be right when the numerator degree exceeds the denominator degree.
Check Your Understanding
For each question, identify the relevant degrees and leading coefficients before deciding the limit.
- Evaluate \(\lim_{n\to\infty}(7n^2-1)/(2n^2+5)\).
- Evaluate \(\lim_{n\to\infty}(n+4)/(3n^3-2)\).
- Does \((4n^3+1)/(2n^2+7)\) approach \(+\infty\) or \(-\infty\)? Explain using the leading coefficients.
- What must be checked before applying the degree rule to a rational sequence?
- For \((n^2-9)/(n-3)\), which index makes the original expression undefined, and what is the limit when the sequence is considered for \(n\ge4\)?
