The sequential criterion for function limits

The condition $x_n \neq x_0$ ensures that the limit describes the behavior of the function *near* $x_0$, independent of the function's value *at* $x_0$. It excludes the trivial case where the sequence is constantly $x_0$, which would always yield $f(x_0)$ regardless of the surrounding behavior.

Conditions: $x_0$ is an accumulation point of the domain.; The sequence $x_n$ approaches $x_0$.; The function $f$ may or may not be defined at $x_0$.

The sequential criterion requires that *every* admissible sequence approaching the accumulation point yields the same limit. Checking only a few paths provides only a subset of cases; a counterexample could exist in an unchecked path, invalidating the universal claim.

Conditions: The criterion applies to all sequences in the domain approaching $x_0$ (excluding $x_0$).; A single counterexample is sufficient to disprove the limit.; Finite verification cannot establish a universal truth for infinite sets.

If the function limit exists, all input sequences approaching $x_0$—including those from the left and right—must produce output sequences converging to the same limit. Discrepancy between left and right limits implies the overall limit does not exist.

Conditions: The accumulation point is $x_0$.; Sequences are chosen from the domain approaching $x_0$ from different directions (e.g., $x < x_0$ and $x > x_0$).; The function limit is assumed to potentially exist.

If two sequences $x_n$ and $y_n$ both approach $x_0$ (with terms not equal to $x_0$) but their function values $f(x_n)$ and $f(y_n)$ approach different limits $A$ and $B$ ($A \neq B$), then the function limit $\lim_{x \to x_0} f(x)$ does not exist.

Conditions: Both sequences must approach the same accumulation point $x_0$.; Neither sequence can contain the point $x_0$ itself.; The limits of the output sequences must be distinct ($A \neq B$).

$x^2$ exhibits stable, predictable convergence where all sequences approaching 0 yield outputs approaching 0. In contrast, $\sin(1/x)$ oscillates infinitely rapidly near 0, allowing the construction of sequences with distinct output limits (e.g., 1 and -1), thus failing the sequential criterion.

Conditions: Both functions are analyzed at the accumulation point $x_0 = 0$.; The comparison focuses on the behavior of $f(x_n)$ for sequences $x_n \to 0$.; $x_n \neq 0$ for all sequences considered.

The function $\sin(1/x)$ has no limit as $x \to 0$ because one can construct two sequences approaching 0 whose function values oscillate between 1 and -1, violating the requirement that all sequences must converge to the same limit.

Conditions: The domain excludes $x=0$ for the limit consideration.; Sequences must consist of points where the function is defined.; The accumulation point is $x_0 = 0$.

The graph of $f(x)=x^2$ shows a smooth, continuous curve passing through the origin. Visually, any sequence of x-values approaching 0 maps to y-values approaching 0, illustrating that the output limit is consistent regardless of the input path.

Conditions: The function is $f(x) = x^2$.; The accumulation point is $x_0 = 0$.; The graph is used as a visual aid, not a formal proof.

The sequential criterion states that the limit of $f(x)$ as $x$ approaches $x_0$ is $A$ if and only if every sequence of domain points $x_n$ approaching $x_0$ (with $x_n \neq x_0$) has its function values $f(x_n)$ approaching $A$.

Conditions: $x_0$ is an accumulation point of the domain.; The sequence $x_n$ must consist of points in the domain.; $x_n \neq x_0$ for all $n$.