The convergence of a sequence to a limit is only useful in practice once it can be computed for sequences built out of simpler ones, by adding, subtracting, multiplying, or dividing them, without having to fall back on the \(\epsilon\)-\(N\) definition every single time. This chapter derives exactly those rules, together with two of the standard tools used to prove the harder limits they can't reach directly: Bernoulli's inequality and the squeeze theorem. The same rules, extended from sequences to functions at the end, are what makes the limit definition of the derivative workable in the first place.
The Algebra of Limits
Let \((a_n)\) and \((b_n)\) be convergent sequences with \(\lim_{n\to\infty}a_n=a\) and \(\lim_{n\to\infty}b_n=b\). Then the sequences obtained by adding, subtracting, scaling, multiplying, and (whenever \(b\neq0\)) dividing them are convergent as well, with
\[ \begin{array}{rl} \lim\limits_{n\to\infty}(a_n+b_n) &= a+b\\ \lim\limits_{n\to\infty}(a_n-b_n) &= a-b\\ \lim\limits_{n\to\infty}(c\cdot a_n) &= c\cdot a, \quad c\in\mathbb{R}\\ \lim\limits_{n\to\infty}(a_n\cdot b_n) &= a\cdot b\\ \lim\limits_{n\to\infty}\dfrac{a_n}{b_n} &= \dfrac{a}{b}, \quad b\neq0. \end{array} \]
Proof of the Sum and Difference Rule
Let \(\epsilon>0\). Since \(a_n\to a\), there is an \(N_1\) with \(|a_n-a|<\epsilon/2\) for all \(n\geq N_1\); since \(b_n\to b\), there is an \(N_2\) with \(|b_n-b|<\epsilon/2\) for all \(n\geq N_2\). For every \(n\geq\max(N_1,N_2)\), the triangle inequality gives
\[ \bigl|(a_n+b_n)-(a+b)\bigr| \leq |a_n-a|+|b_n-b| < \frac{\epsilon}{2}+\frac{\epsilon}{2} = \epsilon, \]
so \(a_n+b_n\to a+b\). The same estimate with a minus sign in front of \(b_n\) and \(b\) proves the difference rule, and setting \(b_n=c\) constant recovers the scalar rule.
Proof of the Product Rule
Since \((a_n)\) converges, it is bounded (every convergent sequence is), so there is an \(M>0\) with \(|a_n|\leq M\) for all \(n\). Inserting and removing \(a_nb\) gives
\[ a_nb_n-ab = a_n(b_n-b) + b(a_n-a), \]
hence \(|a_nb_n-ab| \leq M|b_n-b| + |b||a_n-a|\). Given \(\epsilon>0\), choose \(N_1\) so that \(|b_n-b|<\epsilon/(2M)\) for \(n\geq N_1\), and \(N_2\) so that \(|a_n-a|<\epsilon/(2(|b|+1))\) for \(n\geq N_2\). For \(n\geq\max(N_1,N_2)\) both terms are below \(\epsilon/2\), so \(a_nb_n\to ab\).
Proof of the Quotient Rule
It suffices to show \(1/b_n\to1/b\) whenever \(b\neq0\); the quotient rule then follows from the product rule applied to \(a_n\cdot(1/b_n)\). Since \(b_n\to b\neq0\), there is an index beyond which \(|b_n-b|<|b|/2\), and the reverse triangle inequality gives \(|b_n|>|b|/2>0\) for those \(n\), so \(1/b_n\) is defined. For such \(n\),
\[ \left|\frac{1}{b_n}-\frac{1}{b}\right| = \frac{|b-b_n|}{|b_n||b|} < \frac{2}{|b|^2}\,|b_n-b|. \]
Given \(\epsilon>0\), choosing \(n\) large enough that \(|b_n-b|<\epsilon|b|^2/2\) makes the right-hand side smaller than \(\epsilon\), so \(1/b_n\to1/b\), and therefore \(a_n/b_n\to a/b\).
Worked Example
Take \(a_n=5+\frac{1}{n^3}\) and \(b_n=2-\frac{3}{n}\). Both perturbation terms are null sequences, so \(a_n\to5\) and \(b_n\to2\), and the algebra of limits gives every combination without any further work:
\[ \lim_{n\to\infty}(a_n+b_n)=7, \quad \lim_{n\to\infty}(a_n-b_n)=3, \quad \lim_{n\to\infty}(a_n\cdot b_n)=10, \quad \lim_{n\to\infty}\frac{a_n}{b_n}=\frac{5}{2}. \]
Limits of Rational Sequences
A ratio of two polynomials in \(n\) can always be reduced to the rules above by dividing numerator and denominator by the highest power of \(n\) that appears anywhere in the fraction. Every term that is left with a positive power of \(n\) in the denominator becomes a null sequence, and only the coefficients of the highest power in the original numerator and denominator survive:
\[ \lim_{n\to\infty}\frac{2n+5}{4n-7} = \lim_{n\to\infty}\frac{2+\frac{5}{n}}{4-\frac{7}{n}} = \frac{2+0}{4-0} = \frac{1}{2}. \]
More generally, if the numerator has degree \(p\) and the denominator has degree \(q\) (both with leading coefficients \(a_p\) and \(b_q\)), dividing through by \(n^{\max(p,q)}\) and applying the algebra of limits shows the ratio converges to \(a_p/b_q\) if \(p=q\), to \(0\) if \(p<q\), and grows without bound if \(p>q\), the same "\(\pm\infty\) is not a valid limit" situation already noted for sequences in general.
Bernoulli's Inequality
Two of the standard limits stated without proof for sequences, \(\lim a^n=0\) for \(|a|<1\) and \(\lim\sqrt[n]{n}=1\), need an estimate that the algebra of limits alone cannot supply. Both follow from Bernoulli's inequality, named after Jakob Bernoulli (1655–1705):
\[ (1+x)^n \geq 1+nx \qquad \text{for every } x\geq-1 \text{ and every } n\in\mathbb{N}_0. \]
The proof is by induction on \(n\). For \(n=0\) both sides equal \(1\). Assuming the inequality holds for some \(n\), and using \(1+x\geq0\) to multiply both sides of \((1+x)^n\geq1+nx\) by \(1+x\) without flipping the inequality,
\[ (1+x)^{n+1} \geq (1+nx)(1+x) = 1+(n+1)x+nx^2 \geq 1+(n+1)x, \]
since \(nx^2\geq0\). This proves the inequality for \(n+1\), completing the induction.
Application: Geometric Null Sequences
For \(q=0\) the claim \(\lim q^n=0\) is trivial. For \(0<|q|<1\), write \(\frac{1}{|q|}=1+a\) with \(a=\frac{1}{|q|}-1>0\). Bernoulli's inequality gives \((1+a)^n\geq1+na\), hence
\[ 0 \leq |q|^n = \frac{1}{(1+a)^n} \leq \frac{1}{1+na}. \]
The right-hand side is a null sequence by the rational-sequence rule above, so \(|q|^n\to0\) by the squeeze theorem proved next, and therefore \(q^n\to0\).
Application: The \(n\)-th Root of \(n\)
Write \(\sqrt[n]{n}=1+h_n\) with \(h_n\geq0\) (since \(n\geq1\) forces \(\sqrt[n]{n}\geq1\)). Raising both sides to the \(n\)-th power and keeping only the quadratic term of the binomial expansion,
\[ n = (1+h_n)^n \geq 1+nh_n+\binom{n}{2}h_n^2 \geq \binom{n}{2}h_n^2 = \frac{n(n-1)}{2}h_n^2, \]
so \(h_n^2\leq\frac{2}{n-1}\) and therefore \(0\leq h_n\leq\sqrt{\frac{2}{n-1}}\). The right-hand bound tends to \(0\), so the squeeze theorem gives \(h_n\to0\), and hence \(\sqrt[n]{n}=1+h_n\to1\).
The Squeeze Theorem
If \(a_n\leq b_n\leq c_n\) for all \(n\) beyond some index, and \((a_n)\) and \((c_n)\) converge to the same limit \(L\), then \((b_n)\) converges to \(L\) too. Given \(\epsilon>0\), \(a_n>L-\epsilon\) and \(c_n<L+\epsilon\) eventually, so
\[ L-\epsilon < a_n \leq b_n \leq c_n < L+\epsilon \]
eventually as well, which is exactly \(|b_n-L|<\epsilon\).
As an example, since \(|\sin n|\leq1\) for every \(n\),
\[ -\frac{n}{n^2+n+1} \leq \frac{n\sin n}{n^2+n+1} \leq \frac{n}{n^2+n+1}. \]
Both bounding sequences have numerator degree \(1\) and denominator degree \(2\), so both tend to \(0\) by the rational-sequence rule, and the squeeze theorem gives \(\lim_{n\to\infty}\frac{n\sin n}{n^2+n+1}=0\).
Limits of Functions
Every rule above transfers to functions once a limit for functions is defined in terms of the sequence limit already available. Following Eduard Heine, for a function \(f\) defined near \(a\) (though not necessarily at \(a\) itself),
\[ \lim_{x\to a}f(x)=L \quad:\Longleftrightarrow\quad f(x_n)\to L \text{ for every sequence } x_n\to a \text{ with } x_n\neq a. \]
Because this reduces the statement to ordinary sequence convergence for every approaching sequence \((x_n)\), the sum, difference, scalar, product, and quotient rules, together with the squeeze theorem, carry over immediately: if \(\lim_{x\to a}f(x)\) and \(\lim_{x\to a}g(x)\) both exist, so does the limit of their sum, product, and (denominator permitting) quotient, with the same formulas proved above for sequences. Limits as \(x\to\infty\) and one-sided limits \(x\to a^{\pm}\) are defined the same way, by restricting to sequences that diverge to \(\infty\) or approach \(a\) from one side only.
This is precisely the notion of limit used in the definition of the derivative, \(f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\): the sum, product, and quotient rules for derivatives are nothing more than the corresponding algebra-of-limits rules above, applied to the difference quotients of two functions instead of to two plain sequences. The same transfer justifies standard function limits such as \(\lim_{h\to0}\frac{\sin h}{h}=1\), used repeatedly when deriving trigonometric derivatives from the limit definition.
Two chapters build directly on this notion of a functional limit: the chapter on continuity turns \(\lim_{x\to a}f(x)=f(a)\) into a whole toolkit (the intermediate and extreme value theorems, uniform continuity, continuous extension), while the chapter on monotone, bounded-variation, and convex functions studies exactly the one-sided limits \(x\to a^{\pm}\) mentioned above, which always exist for monotone functions even where the two-sided limit does not.
Landau Notation: Comparing the Growth of Functions
Limits also make precise what it means for one function to grow no faster, strictly slower, or at the same rate as another, near a point \(a\) (finite or \(\pm\infty\)) — the Landau symbols, named after Edmund Landau. Throughout, \(g(x)\neq0\) near \(a\) (excluding \(a\) itself).
\[ \begin{array}{rl} f(x)=O\bigl(g(x)\bigr) \text{ as } x\to a &:\Longleftrightarrow\quad \limsup_{x\to a}\left|\dfrac{f(x)}{g(x)}\right| < \infty,\\[0.4em] f(x)=o\bigl(g(x)\bigr) \text{ as } x\to a &:\Longleftrightarrow\quad \lim_{x\to a}\dfrac{f(x)}{g(x)} = 0. \end{array} \]
Equivalently, and closer to how the definitions are used in practice, \(f=O(g)\) as \(x\to a\) means there are constants \(C>0\) and \(\delta>0\) (or, if \(a=\infty\), a bound \(R\)) with
\[ |f(x)| \leq C\,|g(x)| \quad \text{whenever } 0<|x-a|<\delta \text{ (or } x>R\text{)}, \]
i.e. \(f\) is eventually bounded by a constant multiple of \(g\); \(f=o(g)\) is the strict version, where that ratio itself vanishes in the limit rather than merely staying bounded. \(\Omega\) and \(\omega\) are the mirror notions with the inequality reversed — \(f=\Omega(g)\) means \(g=O(f)\), and \(f=\omega(g)\) means \(g=o(f)\) — and \(f=\Theta(g)\) means both \(f=O(g)\) and \(f=\Omega(g)\) hold simultaneously, i.e. \(f\) and \(g\) grow at exactly the same rate up to constants:
\[ f(x)=\Theta\bigl(g(x)\bigr) \text{ as } x\to a \quad:\Longleftrightarrow\quad 0 < \liminf_{x\to a}\left|\frac{f(x)}{g(x)}\right| \leq \limsup_{x\to a}\left|\frac{f(x)}{g(x)}\right| < \infty. \]
Two points are worth flagging about the notation itself. First, \(O(g)\) really denotes the set of all functions eventually bounded by a multiple of \(g\); writing \(f=O(g)\) is a standard abuse of the equals sign for \(f\in O(g)\), and it is not symmetric — \(x=O(x^2)\) as \(x\to\infty\) does not make \(O(x^2)=x\) meaningful. Second, the point \(a\) matters: \(x^2=O(x)\) is false as \(x\to\infty\) but true as \(x\to0\), since boundedness of \(|x^2/x|=|x|\) depends entirely on which limit is being taken.
Growth Hierarchy: Polynomials, Logarithms, and Exponentials
As \(x\to\infty\), \(x^p=O(x^q)\) exactly when \(p\leq q\), matching the rational-sequence rule above applied to \(x^p/x^q=x^{p-q}\): the ratio stays bounded (indeed tends to \(0\) or a constant) when \(p\leq q\), and is unbounded when \(p>q\). Two standard comparisons go beyond polynomials entirely and are used constantly:
\[ \ln x = o(x^\varepsilon) \text{ as } x\to\infty \text{ for every } \varepsilon>0, \qquad x^n = o(e^x) \text{ as } x\to\infty \text{ for every } n\in\mathbb{N}. \]
In words: every positive power of \(x\) eventually outgrows every power of \(\ln x\), and \(e^x\) eventually outgrows every polynomial, no matter how large a fixed degree \(n\) or how small a fixed exponent \(\varepsilon\) is chosen.
Application: Running Time of Algorithms
Landau notation is the standard language for running time, where \(x\) is replaced by an input size \(n\to\infty\). Evaluating a degree-\(n\) polynomial by Horner's method takes \(O(n)\) multiplications; binary search takes \(O(\log n)\) comparisons; and the bound \(O(n)\) is tight in both cases in the sense of \(\Theta(n)\), since neither algorithm can do better than a constant multiple of its stated bound on every input. The little-o notation captures a derivative directly, too: \(f\) is differentiable at \(x_0\) with derivative \(f'(x_0)\) exactly when
\[ f(x_0+h) = f(x_0) + f'(x_0)h + o(h) \quad \text{as } h\to0, \]
which is just the limit definition of the derivative rewritten to say the linear approximation's error vanishes strictly faster than \(h\) itself.