FALSE: $\lim_{x \to c} f(g(x)) = M$ whenever $\lim_{x \to c} g = L$ and $\lim_{y \to L} f = M$

False claim: let $A, B \subseteq \mathbb{R}$, let $g : A \to \mathbb{R}$ with $g(A) \subseteq B$ and $f : B \to \mathbb{R}$, let $c$ be a limit point of $A$ and $L$ a limit point of $B$. If

$\lim_{x \to c} g(x) = L \qquad \text{and} \qquad \lim_{y \to L} f(y) = M ,$

then the limit of $f \circ g$ at $c$ exists and $\lim_{x \to c} f\bigl(g(x)\bigr) = M$ (The $\varepsilon$-$\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ at a limit point $c$ of $A$).

This is the statement of Composition of limits holds under either hypothesis: $f$ is defined at $L$ with value $M$, or $g$ avoids $L$ on a punctured neighbourhood of $c$ with both of its extra hypotheses removed, and it is false. It is refuted below by a pair in which $g$ is constant and $f$ has a removable defect at the value of that constant.

Where the naive argument breaks. The inner limit gives $|g(x) - L| < \rho$ for $x$ near $c$; the outer limit gives $|f(y) - M| < \varepsilon$ for $y \in B$ with $0 < |y - L| < \rho$. To combine them at $y = g(x)$ one needs $|g(x) - L| > 0$, and nothing in the hypotheses supplies that. Where $g(x) = L$, the only information available about $f$ is its value $f(L)$, and The $\varepsilon$-$\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ at a limit point $c$ of $A$ says nothing whatever about that value (FALSE: $\lim_{x \to c} f(x) = f(c)$ whenever both sides exist). The two hypotheses of Composition of limits holds under either hypothesis: $f$ is defined at $L$ with value $M$, or $g$ avoids $L$ on a punctured neighbourhood of $c$ are exactly the two ways of closing that gap.

• The failure is not an artefact of the constant inner function. What matters is that $g$ takes the value $L$ on every punctured neighbourhood of $c$; a non-constant $g$ that hits $L$ along a sequence tending to $c$ would fail in the same way. Conversely, replacing $g$ by the identity — which avoids the value $0$ off the point $0$ itself — restores the conclusion, and the companion page carries out that comparison.

• Textbook statements almost always assume continuity of the outer function, which is hypothesis (i) of Composition of limits holds under either hypothesis: $f$ is defined at $L$ with value $M$, or $g$ avoids $L$ on a punctured neighbourhood of $c$ written out. The version with hypothesis (ii) is the one that licenses substitutions such as $y = 1/x$, where the inner function omits the critical value for a structural reason.

• The same witness refutes nothing else on this page. In particular it does not bear on Sums, scalar multiples, products and quotients of function limits, the quotient under the hypothesis that the denominator limit is nonzero: sums, products and quotients are formed pointwise from the values of $f$ and $g$ at the same argument, and no composition is involved.