Free · Year 5 of high school · 50 minutes · 15 questions · no penalty

Practice maths class test — Year 5: limits and continuity.

15 questions drawn at random from a bank of 25, on the block of the programme your teacher is finishing: every attempt is a different test. No penalty for wrong answers, so answer even if you are not sure. At the end you receive by email the correct answer to every question and your weak topics — with an account you see them straight away here on the page. The step-by-step solution unlocks with your first lesson.

A class test is the written test an Italian high-school class sits after each block of the programme; this practice version is multiple choice.

Test in ItalianThe questions of this simulator are in Italian, as in the real exam.

Test prepared by Gaetano Livornese · Aerospace engineer, Politecnico di Milano · maths and physics tutor · updated 13 October 2026

Programme covered by the test

Algebra of limits and indeterminate forms
Standard limits
Continuity at a point and on an interval
Discontinuities of the first, second and third kind
Vertical, horizontal and oblique asymptotes
Sketching the graph of a function

The test on limits opens Year 5, in September-October, and for many it is the first heavy grade of the approaching maturità: indeterminate forms, at least two standard limits, the classification of discontinuities and the asymptotes of a function with fractions, with a sketch of the graph to draw at the end.

The mistake that comes back most often is about standard limits applied without first bringing the argument into the right form: the limit of sin(5x)/x as x tends to 0 is not 1. You need to multiply and divide by 5, writing 5 · sin(5x)/(5x), which tends to 5 · 1 = 5. The standard limit only holds when the argument of the sine matches exactly what is in the denominator.

The second is about the continuity of piecewise functions: checking only that the two “pieces” give the same value at the joining point is not enough. The limit from the left, the limit from the right and the value of the function at that point must be calculated separately and all three compared — skipping one of the three is the mistake that goes unnoticed until the marking arrives.

The third is about 0/0 forms with a difference of roots, for example (√x − 2)/(x − 4) as x tends to 4: trying to simplify directly gets you nowhere, because the indeterminate form remains. You need to multiply by the conjugate, √x + 2, which turns the numerator into x − 4 and lets you cancel it with the denominator: the result is 1/(√x + 2), that is 1/4.

To prepare, redo the standard limits with an argument other than x (5x, x², 1/x) before the practice test: that is where most marks are lost, not in the rule itself.

Two worked examples from the test

Question 1 · Limiti, forme indeterminate, continuità, asintoti · Limiti e forme indeterminate

Calcola $x \to + \infty lim \frac{3 x ^{2} - 5 x + 1}{2 x ^{2} + 7} .$

A $0$
B $\frac{3}{2}$ correct answer
C $\frac{1}{7}$
D $+ \infty$

Solution

Passo 1 — Riconosco la forma. Numeratore e denominatore tendono entrambi a $+ \infty$ : forma indeterminata $\frac{\infty}{\infty}$ .

Passo 2 — Raccolgo il termine di grado massimo. Il grado massimo è $2$ sia sopra sia sotto:

$\frac{x ^{2} ( 3 - \frac{5}{x} + \frac{1}{x ^{2}} )}{x ^{2} ( 2 + \frac{7}{x ^{2}} )} = \frac{3 - \frac{5}{x} + \frac{1}{x ^{2}}}{2 + \frac{7}{x ^{2}}}$

Passo 3 — Passo al limite. Tutti i termini con $x$ al denominatore tendono a $0$ :

$x \to + \infty lim \frac{3 - 0 + 0}{2 + 0} = \frac{3}{2}$

Quando i gradi sono uguali il limite è il rapporto dei coefficienti direttivi, non dei termini noti: se avevi scelto $\frac{1}{7}$ hai guardato $1$ e $7$ , cioè proprio i termini noti. Se avevi scelto $0$ hai pensato che i due $x^{2}$ "si semplificassero" facendo sparire tutto.

Question 2 · Limiti, forme indeterminate, continuità, asintoti · Limiti e forme indeterminate

Calcola $x \to + \infty lim (x^{2} + 6 x - x) .$

A $0$
B $6$
C $+ \infty$
D $3$ correct answer

Solution

Passo 1 — Riconosco la forma. Entrambi i termini tendono a $+ \infty$ : forma indeterminata $\infty - \infty$ .

Passo 2 — Razionalizzo moltiplicando e dividendo per l'espressione coniugata $x^{2} + 6 x + x$ :

$\frac{( x ^{2} + 6 x ) - x ^{2}}{x ^{2} + 6 x + x} = \frac{6 x}{x ^{2} + 6 x + x}$

Passo 3 — Raccolgo $x$ , che per $x \to + \infty$ è positivo, quindi $x^{2} + 6 x = x 1 + \frac{6}{x} :$

$\frac{6 x}{x ( 1 + \frac{6}{x} + 1 )} = \frac{6}{1 + \frac{6}{x} + 1}$

Passo 4 — Passo al limite.

$\frac{6}{1 + 1} = 3$

$\infty - \infty$ non fa $0$ : i due infiniti crescono quasi allo stesso modo e la differenza resta finita. Se hai risposto $6$ ti sei fermato al numeratore dimenticando che il denominatore tende a $2$ , non a $1$ .

In the online practice test you will find 13 more questions like these; the correct answer to every question arrives by email.

Prefer paper? Download the test as a PDF (15 questions, 5 pages · 86 KB, ready to print). You receive the solutions by email after taking it online.

Download the PDF

What a maths test on limits and continuity looks like

Limits with indeterminate forms of the types ∞/∞, 0/0 and ∞ − ∞, at least two standard limits, the classification of the points of discontinuity, the asymptotes of a function with fractions and usually a graph to sketch.

The mistake that costs the grade

Treating ∞ as a number and simplifying ∞ − ∞ to zero. On asymptotes, looking for the oblique one when the horizontal one already exists: it is one or the other, never both on the same side, and looking for both wastes time and marks.

How to use this practice test

Take it once cold, without revising and with the timer on: it is there to show you where you stand, not to look good. Then look at your weak topics, revise only those — usually two or three, not the whole chapter — and retake it after a few days. The 15 questions are drawn from a bank of 25, so the second attempt is not the same test and the scores are genuinely comparable.

Is the score not enough?

See which topics lost you the most marks, then read the guide to limits explained well — or we look at your results together: the diagnostic is free.

Book the free diagnostic call The other Year 5 tests