The limit is not the function value: where the function is heading, not where it arrives

01The mistake

Students evaluate every limit by substitution. When the function has a hole at $x=2$ but approaches 5 there, they report the limit as undefined, because $f(2)$ is undefined. The limit is 5. The value at the point is the one thing the limit is defined to ignore.

The tell is a piecewise function with a value defined away from the trend. If $f(x) = x+1$ for $x \neq 3$ and $f(3) = 10$, then $\lim_{x\to 3} f(x) = 4$ while $f(3) = 10$. Students who answer 10 have merged the two ideas. This item takes fifteen seconds and separates the class better than any amount of algebraic limit practice.

It hollows out continuity. Continuity at $a$ is the statement that $\lim_{x\to a} f(x) = f(a)$ — which is only worth stating if the two can differ. A student for whom they are the same by definition experiences continuity as a tautology, cannot see what a removable discontinuity removes, and has no way to understand why the definition has three separate conditions.

Tall and Vinner (1981) framed this as a gap between concept definition and concept image: students hold a formal definition alongside a mental picture that may contradict it, and the picture is what they reason with. They found that the informal phrasing — getting as close as we please — leads many students to conclude that a function cannot reach its limit, which is the same fusion running in the opposite direction.

02Why it makes sense to the student

Substitution works nearly always, and it is taught first. Every polynomial, every rational function away from its zeros, every trig function on its domain: plug in the number and read the answer. Students accumulate hundreds of successful substitutions before meeting a case where it fails.

The functions they met before calculus were continuous everywhere they were defined. Holes, jumps and removable discontinuities are calculus objects. There was no prior experience in which the value and the trend could disagree, so no reason to hold them as separate ideas.

The notation is compact and easy to read as an instruction. $\lim_{x\to a}$ can be read as “put $a$ in,” and for most problems that reading produces the right answer. A wrong rule that succeeds is indistinguishable from a right one until a counterexample arrives.

And the arrow in $x \to a$ genuinely suggests arrival. Students read it as $x$ becoming $a$, rather than as $x$ getting arbitrarily close without the value at $a$ mattering. The symbol invites the merge.

03The correction

State the exclusion as the point of the definition, not as a caveat: the limit as $x \to a$ is determined entirely by values of $f$ at $x \neq a$. What happens at $a$ is irrelevant to it. Say this before any evaluation technique, because a student who learns substitution first will never revisit it.

Then build the three-way example and keep returning to it. Take $f(x) = \frac{x^2-4}{x-2}$, which simplifies to $x+2$ for all $x \neq 2$ and is undefined at 2. The limit as $x\to 2$ is 4; $f(2)$ does not exist. Now define $g$ identically except $g(2) = 100$. The limit is still 4. Changing the value at the point changed nothing, because the limit never consulted it.

Use a table to make the approach visible. Evaluate at $1.9, 1.99, 1.999$ and $2.1, 2.01, 2.001$, and let students watch the outputs converge on 4 while the row at exactly 2 stays empty. The empty row is the lesson — it is the one input the limit does not use.

Then continuity becomes a real claim rather than a definition to recite. $f$ is continuous at $a$ when the limit exists, $f(a)$ exists, and the two are equal — three conditions, because there are three ways to fail. Students who have seen the limit and the value disagree can generate all three failures themselves.

A useful classroom test: “$f(x) = x + 1$ for all $x \neq 3$, and $f(3) = 10$. What is $\lim_{x\to 3} f(x)$, and what is $f(3)$?” Two questions, two different answers, and a student who cannot produce two different answers has the two concepts fused whatever their limit algebra looks like.

04A sample question

05What each wrong answer reveals

• A Limit read as the value. The student substituted, which is the method that has worked on nearly every limit they have evaluated. Notice they are not wrong about $f(3)$ — it genuinely is 10. They answered a question that was not asked, because in their concept image the two questions are the same one. The table of approaching values is the correction, and the empty row at $x=3$ is the part that does the work.
• B Correct. As $x$ approaches 3 from either side, $f(x) = x+1$ approaches 4. The value $f(3) = 10$ is irrelevant to the limit, which depends only on values at $x \neq 3$.
• C Discontinuity read as non-existence. The student has correctly identified that the function is discontinuous at 3 — a real observation — and concluded that the limit therefore fails to exist. This is the reverse fusion: they know the limit and value differ and have taken that as disqualifying. The repair is that a removable discontinuity is defined by the limit existing while differing from the value, which is worth stating as the whole point of the word “removable.”
• D An average invented to reconcile them. Rare, and revealing. This student sees two candidate answers, cannot choose, and splits the difference. It indicates no model of what a limit is rather than a wrong one, and it needs the definition built from the table rather than any correction to their reasoning.

A and C are the same fusion resolved in opposite directions: A keeps the value, C throws both out. C is closer, because that student has noticed the discrepancy and only needs to learn that a limit can exist where a function is discontinuous. D has no model. Only B separates approach from arrival, and the two-question test above is the fastest way to find out which of these a student holds before the unit gets any further.

06Try it in Mistake Master