Sequential Characterization of Limits

The sequential characterization is the standard tool for proving nonexistence of limits in every introductory real analysis course, and it reappears immediately in the definition of Continuity via sequences (continuity of $f$ at $c$ becomes: $x_n \to c \implies f(x_n) \to f(c)$, with no need to exclude $x_n = c$ this time, since $f(c)$ is now part of the answer). It underlies rigorous treatments of the derivative as a limit of difference quotients, where sequential arguments are often cleaner than direct $\varepsilon$-$\delta$ estimates. In numerical analysis and scientific computing, iterative algorithms that generate a discrete sequence of approximations $x_n \to c$ rely implicitly on this correspondence to justify that computed sequential limits track the true continuous limit. In topology, the same idea generalizes to nets (for general topological spaces, where sequences alone may not suffice) and to filters, both built explicitly to repair the sequential criterion in settings without countable neighborhood bases. In complex analysis and functional analysis, sequential compactness arguments and continuity of operators between normed spaces routinely invoke this same equivalence to reduce continuum-indexed statements to countable, discrete ones. The idea also underlies the "sandwich" or squeeze techniques taught alongside The Algebra of Limits, since squeezing is often first proved for sequences and then ported to functions via exactly this correspondence.

Ask the tutor: How does this sequential criterion change when we define continuity instead of a general limit, and why does the exclusion $x_n \neq c$ disappear there?

Think of $c$ as a target on the number line, and imagine every conceivable way of walking toward it without ever landing exactly on it: from the left, from the right, hopping around erratically, spiraling in with alternating signs, whatever you like — as long as the walk converges to $c$. The $\varepsilon$-$\delta$ definition says $f(x) \to L$ when $x$ is continuously close to $c$. The sequential criterion says instead: no matter which of these discrete walks $(x_n)$ toward $c$ you choose, the corresponding heights $f(x_n)$ must all funnel toward the same value $L$. The power of the equivalence is that it lets you swap perspectives. The $\varepsilon$-$\delta$ side is a universal statement about a continuum (hard to falsify directly, since there are uncountably many $x$ near $c$ to check). The sequential side is a universal statement about sequences, but it becomes easy to falsify: you just need one bad walk. If $f(x_n) \to L_1$ along one path and $f(y_n) \to L_2 \neq L_1$ along another, the sequential criterion fails for every $L$, so — by the theorem — the $\varepsilon$-$\delta$ limit does not exist either. The figure below shows this "many paths, one destination" picture for a function with a genuine limit at $c$, alongside a warning sign for a function like $\sin(1/x)$ where different paths disagree.

Example 1 (confirming a limit via the sequential criterion, asymmetric case). Let $f(x) = \dfrac{x^2 - 5x + 6}{x - 2}$ for $x \neq 2$ (undefined at $x=2$, note $D = \mathbb{R}\setminus\{2\}$), and let $c = 2$. We claim $\lim_{x\to 2} f(x) = -1$. First simplify algebraically for $x \neq 2$:

This is valid precisely because we divide by $x - 2 \neq 0$, which holds for every $x \in D$. We verify the sequential criterion directly (bypassing $\varepsilon$-$\delta$, since the theorem lets us). Let $(x_n)$ be any sequence with $x_n \in D \setminus \{2\}$ (i.e. $x_n \neq 2$ for all $n$, automatically true since $2 \notin D$) and $x_n \to 2$. Then for every $n$,

By the algebra of limits for sequences (subtraction of a convergent sequence and a constant sequence), since $x_n \to 2$,

This holds for every admissible sequence $(x_n)$, with the same limit $-1$ each time. By the theorem (Direction 2, applied in reverse as a sufficient condition), we conclude

Notice the asymmetry deliberately built in: the function is a genuinely non-symmetric rational expression, not a disguised even/odd function, and the point $c = 2$ is a removable singularity rather than an endpoint or a "nice" value like $0$.

Example 2 (refuting a limit — a genuinely two-sided oscillatory case). Let

We show $\lim_{x\to 0} g(x)$ does not exist, using the Corollary above with two explicitly asymmetric sequences (not simply $\pm$ mirror images). Define

Both are strictly positive, so $x_n, y_n \in D \setminus \{0\}$ for all $n$. Convergence to $0$: as $n \to \infty$, the denominators $2\pi n + \pi/2$ and $2\pi n + 7\pi/6$ both tend to $+\infty$, so $x_n \to 0$ and $y_n \to 0$ (reciprocal of a sequence diverging to $+\infty$ tends to $0$, a standard fact from the algebra of sequence limits). Evaluating $g$ along $(x_n)$:

using periodicity of $\sin$ with period $2\pi$. So $g(x_n) = 1$ for all $n$, hence $g(x_n) \to 1$. Evaluating $g$ along $(y_n)$:

again by periodicity, and using the exact value $\sin(7\pi/6) = -\sin(\pi/6) = -1/2$. So $g(y_n) \to -\tfrac12$. We have exhibited $x_n \to 0$, $y_n \to 0$, both in $D \setminus\{0\}$, with

By the Corollary, $\lim_{x\to 0}\sin(1/x)$ does not exist. Notice the choice of the second sequence's phase ($7\pi/6$, giving $-1/2$) rather than the more common $-1$ or $0$: any two distinct output values reachable infinitely often suffice, and this shows the argument does not depend on cherry-picking extreme values like the sequence hitting $\pm 1$.