Maths in Year 5 of high school: functions and limits

Mid-September 2026, school back for two weeks: in Year 5 at the languages high school (liceo linguistico) the teacher is doing functions these days — domain, sign, symmetry. The first real test arrives in six to eight weeks, between the end of October and mid-November, and in November-December you move on to limits. There is time, and it is exactly the moment when it is worth using it.

One thing matters more than any exercise: at the languages high school the Year 5 syllabus is, on paper, the same as at the science high school — functions, limits, then derivatives. What changes is the hours, 2 a week against 4, and the level, that of the less demanding textbook. Whoever studies from the science high school textbook gets frightened by things that will never come up in their test — proofs, epsilon-delta, complicated irrational functions. And whoever instead skips the domain thinking that “it’s only languages anyway” gets a real failing grade, because the teacher always asks for that, regardless of the hours.

What really comes up in the functions and limits test in Year 5 at the languages high school

With 2 hours a week the syllabus isn’t compressed by cutting topics at random: it is compressed by staying on rational fractional functions and leaving out everything that serves only to “see the general case”. What the teacher asks, almost always, are these five blocks, in the order you see them in class:

1. the domain of a rational fractional function — the condition of existence of the denominator, no complicated roots;
2. symmetries — checking whether $f(x)$ is even, odd, or neither;
3. intercepts with the axes — where the graph meets $x=0$ and $y=0$;
4. sign — where the function is positive and where it is negative, with the sign table;
5. limits at the ends of the domain, that is at infinity and at the excluded points, to find the asymptotes (vertical, horizontal; the oblique one you only see if the teacher has time, but it is rare at the languages high school with 2 hours).

The typical format is “complete study of a rational function as the standard task”: they give you an $f(x)$ and ask you to do all five steps, with the probable graph at the end. There is no integral, no second derivative, no study of inflection points — that, if it comes, comes later and with a separate test.

What does not come up, and on which it isn’t worth spending an extra hour of study: the proofs of the theorems on limits (Weierstrass, sign permanence, comparison), the formal definition with $\epsilon$ and $\delta$, functions with intertwined roots and absolute values, integrals. The national curriculum guidelines (Indicazioni Nazionali) for the languages high school say so explicitly: in-depth training in limit-calculation techniques is not required. The goal, when you get to derivatives in December-January, is to understand that the derivative is a slope, a speed, and to use it to optimise something — not to train on a hundred differentiation exercises as at the science high school. The same logic applies to potential and field: at the languages high school the idea is enough, without the heavy formalism.

A note that is worth having clear from the start: in 2026 maths is not among the subjects of the maturità oral exam (Italy’s school-leaving exam) at the languages high school. The Year 5 grade weighs on the school credit, not on the exam. This doesn’t mean “you can ignore it” — it still weighs on the school report and on the final grade — but it removes the anxiety of “I have to be ready for the maths oral”: there isn’t one.

The mistakes that cost marks

Almost all the mistakes in the test come from rushing the first two steps, not from the limits themselves.

• Condition of existence of the denominator forgotten. Writing $f(x)=\frac{x+2}{x^2-4}$ and going straight to the sign without ever having written $x\ne \pm 2$. The teacher counts it as a serious mistake even if the rest is right, because without the domain everything that follows is formally incorrect.
• Wrong sign table. You factorise numerator and denominator well but then assign the signs to the intervals without checking the roots in increasing order, or forget that a factor in the denominator, if negative, flips the sign of the fraction in that interval.
• Symmetry tested badly. You calculate $f(-x)$ but don’t simplify all the way before comparing with $f(x)$ and $-f(x)$: an even function looks like “neither” only because you stopped one step too soon.
• Confusing the domain with a “root condition” when there is no root. It happens when you study from the science high school textbook, full of irrational functions: you end up looking for conditions that your function, being only fractional, doesn’t have.
• Vertical asymptote announced where the factor cancels. It is the most insidious mistake and you’ll see it below in the third exercise: if numerator and denominator share a factor, that point is a hole in the graph (a removable discontinuity), not an asymptote. Declaring an asymptote there is a conceptual mistake, not a calculation one, and the teacher weighs it as such.

The method step by step

The sequence is always the same, and it is worth following it in order because each step uses the result of the previous one:

1. Domain. Set the denominator different from zero, solve. It is the basis of everything: a mistake here propagates into every following step.
2. Symmetries. Calculate $f(-x)$ and compare it with $f(x)$: if they coincide the function is even (graph symmetric about the $y$ axis), if $f(-x)=-f(x)$ it is odd (symmetric about the origin). It serves to halve the work on sign and limits, because it is enough to study half the domain and reflect.
3. Intercepts with the axes. Set $x=0$ for the intercept with the $y$ axis (if $x=0$ is in the domain), set $f(x)=0$ for the intercepts with the $x$ axis.
4. Sign. Study where numerator and denominator are positive or negative separately, then combine with the sign table.
5. Limits at the ends of the domain. Calculate the limit for $x\to \pm \infty$ and at the points excluded from the domain, both from the right and from the left.
6. Asymptotes. From the limits in step 5 read off the vertical and horizontal ones: infinite limit at an excluded point → vertical asymptote; finite limit at infinity → horizontal asymptote.

The step the teacher weighs most, almost always, is the fifth: not so much the calculation of the limit itself, as the ability to connect it to the right step — understanding which points have to be checked (the ends of the domain found in step 1, not random points) and what the result says about the graph. A test that gets one calculation wrong but sets up the limits at the right points gets more marks than one that calculates everything correctly in the wrong place.

Three worked exercises

1. $f(x)=\frac{x^2-4}{x^2-9}$

Domain. $x^2-9\ne 0\Rightarrow x\ne \pm 3$.

Symmetry.

Even function, graph symmetric about the $y$ axis.

Intercepts. With the $y$ axis: $f(0)=\frac{-4}{-9}=\frac{4}{9}$. With the $x$ axis: $x^2-4=0\Rightarrow x=\pm 2$.

Sign. Numerator $x^2-4\ge 0$ for $x\le -2$ or $x\ge 2$. Denominator $x^2-9>0$ for $x<-3$ or $x>3$. Combining: $f(x)>0$ for $x<-3$, $f(x)<0$ for $-3<x<-2$, $f(x)>0$ for $-2<x<2$, $f(x)<0$ for $2<x<3$, $f(x)>0$ for $x>3$.

Limits and asymptotes. At the ends of the domain:

and by symmetry the same behaviour, reversed, at $x=-3$: two vertical asymptotes, $x=-3$ and $x=3$. At infinity, degree of numerator equal to degree of denominator, so the limit is the ratio of the leading coefficients:

Horizontal asymptote $y=1$.

If you want to practise on other questions of this same slant, with instant marking, there is the Year 5 practice test on limits: the correct answer to every question is free, and the guided step-by-step solution unlocks with your first lesson.

2. $f(x)=\frac{x}{x^2-1}$

Domain. $x^2-1\ne 0\Rightarrow x\ne \pm 1$.

Symmetry.

Odd function, graph symmetric about the origin.

Intercepts. $f(0)=0$: the only intercept, with both axes, is the origin $(0,0)$.

Sign. Numerator: $x>0$ for $x>0$. Denominator: $(x-1)(x+1)>0$ for $x<-1$ or $x>1$. Combining: $f(x)<0$ for $x<-1$, $f(x)>0$ for $-1<x<0$, $f(x)<0$ for $0<x<1$, $f(x)>0$ for $x>1$.

Limits and asymptotes.

Two vertical asymptotes, $x=-1$ and $x=1$. At infinity, degree of numerator (1) lower than degree of denominator (2):

Horizontal asymptote $y=0$, that is the $x$ axis itself.

3. $f(x)=\frac{x^2-1}{x-1}$ — the trap

Domain. $x-1\ne 0\Rightarrow x\ne 1$. Stopping here, before factorising, is important: the domain is read off the original expression, not the simplified one.

Factorising the numerator:

The graph is the line $y=x+1$, with a hole at the point $x=1$.

Intercepts. With the $y$ axis: $f(0)=1$. With the $x$ axis: $x+1=0\Rightarrow x=-1$ (acceptable, because $-1\ne 1$).

Sign. $f(x)>0$ for $x>-1$ (excluding $x=1$), $f(x)<0$ for $x<-1$.

Limit at the excluded point — here is the trap.

The limit exists and is finite. It is not a vertical asymptote: it is a removable discontinuity (a “third kind” discontinuity in Italian terminology) — a hole in the graph at the point $(1,2)$, where the function is not defined but the limit is. Whoever sees $x=1$ excluded from the domain and writes “vertical asymptote” without checking the limit loses the most insidious point in the test, because the calculation is easy but the conclusion is the wrong one.

At infinity there is nothing to look for: the graph is already a line, apart from the hole, so there is no horizontal asymptote and the oblique line it tends to is the function itself.

If you want to see the same method applied more rigorously, at science high school level — more complex conditions of existence, irrational functions, the formalism we have left out here — there is the guide to domain and symmetries, the first chapter of a complete step-by-step series on studying a function.

How we work in lessons

With 2 hours a week there is no margin to arrive at the test and discover there that the domain isn’t clear: I check it beforehand, with targeted exercises on the rational fractional functions that really come up in your test, not on the whole science high school repertoire. We start from where you are — often it is precisely domain and symmetries, in these weeks — and arrive at limits with the same method you have seen here, repeated until it becomes automatic.

The most convenient starting point is the Year 5 practice test on limits: you do it in a quarter of an hour and it tells you at once where you stand, without anyone asking you anything. From there, if you’d like to try how I work, there is the free 30-minute introductory call, as well as the single lesson and the 10-hour package — the rate overview is on the pricing page (payment offline: bank transfer, Satispay, PayPal, card, Apple Pay, Revolut). The full picture of how I follow the subject is on the maths page. To book the call, or to arrange the first lesson, write to me from contact.