Limits, continuity and asymptotes: the Year 5 test

The test on limits is the first of the final year and, in most classes, the one with the most failing grades of the whole five years. At the liceo scientifico (science-oriented high school) it falls between the end of September and November; in schools with two hours of maths — human sciences, languages, art — it comes later, between November and January, but the test is the same.

The reason it goes badly is not the concept of limit, which almost everyone grasps. It is that the test asks for three different things together — calculating limits, classifying discontinuities, finding asymptotes — and the last two arrive in the last week of lessons, when attention has already gone to calculation. Anyone who arrives at the test knowing only how to calculate gets a 5, even doing all the limits right.

Here you will find the techniques you really need, how discontinuities are classified, the method for asymptotes (oblique included), three worked exercises and the five mistakes that cost the most marks.

What the Year 5 test asks

It is worth clarifying the perimeter at once, because online you mostly find university material and that is something else. On the concept of limit, the definition with $\epsilon$ and $\delta$ and the standard limits seen in depth I have written a separate guide, which also covers the Calculus 1 level: limits explained well. This page is instead calibrated on the Year 5 test, which asks for three blocks:

1. calculating limits with indeterminate forms, without L’Hôpital’s rule — derivatives come later, and using it before the teacher has introduced it is a pointless risk;
2. discussing the continuity of a function and classifying the points of discontinuity;
3. finding the asymptotes of a function, oblique included, which is the exercise worth the most.

The third block is the one on which the grade is decided, and not by chance it is also the bridge to the function study of February.

If your class is already anticipating the theorems of Weierstrass, of zeros and of intermediate values, you will find them stated with their hypotheses in the Year 5 formula sheet, together with all seven indeterminate forms in full and twelve exercises with the complete solution — here we stay within the perimeter of the test.

Indeterminate forms and the three high-school techniques

An indeterminate form is neither a mistake nor a result: it is a signal that direct substitution is not enough and the expression has to be rewritten. The forms that come up in the test are $\frac{0}{0}$, $\frac{\infty }{\infty }$, $\infty -\infty$ and, more rarely, $0\cdot \infty$ and $1^{\infty }$.

There are three techniques, and you choose by looking at the form.

Factoring out the highest-degree term, for $\frac{\infty }{\infty }$ with rational functions. In practice the working rule is the comparison between the degrees of numerator and denominator:

Factorising, for $\frac{0}{0}$ with rational functions. If substituting $x_0$ makes both numerator and denominator vanish, then $(x-x_0)$ is a factor of both: you factorise, cancel, substitute again. It never fails, and that is why the factorisation of Year 1 stays useful until the last day of high school.

Rationalising, for $\infty -\infty$ and for $\frac{0}{0}$ with roots. You multiply and divide by the conjugate expression: the difference of squares makes the root disappear, and the indeterminate form with it.

To these are added the standard limits, which must be known by heart:

There is just one rule for using them and it is often broken: the argument of the function and the denominator must be the same thing. In $\frac{\sin 3x}{x}$ they are not equal, so you adjust:

Continuity and the three types of discontinuity

A function is continuous at $x_0$ if three conditions hold together: $f(x_0)$ exists, the limit as $x\to x_0$ exists and is finite, and the two coincide. When something jumps, the discontinuity is classified like this:

The calculation to do is always the same: right limit, left limit, value of the function. Three numbers, and the classification follows by itself. The jump of the first kind also has a size, that is the difference between the two limits, and teachers often ask for it.

A practical note: piecewise-defined functions are the favourite case of tests, because they allow the question “for what value of the parameter $k$ is the function continuous?”. It is solved by requiring that the right and left limits at the joining point be equal.

Asymptotes: the method, in order

Vertical. They are sought at the points excluded from the domain. The line $x=c$ is a vertical asymptote if

Both one-sided limits must be calculated, because the behaviour on the right and on the left can differ — and that is exactly what the test wants to see written down.

Horizontal. The line $y=L$ is a horizontal asymptote if

Here too the two sides must be looked at separately: a function can have a horizontal asymptote on the right and a different one (or none) on the left.

Oblique. They are sought only if on that side there is no horizontal asymptote: the two exclude each other. The line is $y=mx+q$ with

and $m$ must be finite and non-zero. For a rational function there is a shortcut worth knowing: the oblique asymptote exists when the degree of the numerator exceeds that of the denominator by exactly one, and it is obtained by polynomial division — the quotient is the asymptote. It is the quickest way to check the result of the two limits.

Three worked exercises

1. Form $\frac{\infty }{\infty }$

Substituting gives $\frac{\infty }{\infty }$. I factor out $x^2$ above and below:

All the terms with $x$ in the denominator tend to zero. The result is the ratio of the leading coefficients, as the table says — but the factoring step must be written down: it is what the teacher assesses.

2. Form $\frac{0}{0}$

Substituting, both numerator and denominator vanish: form $\frac{0}{0}$, so $(x-2)$ is a common factor. I factorise:

The limit exists and is finite but the function at $x=2$ is not defined: it is a discontinuity of the third kind, removable. If the question asks for the classification, the answer is already here.

3. Asymptotes of a rational function

Find the domain and asymptotes of $f(x)=\frac{x^2+1}{x-1}$.

The domain is $x\ne 1$. I begin with the vertical asymptote, with the two one-sided limits:

(the numerator tends to $2$, positive; the denominator tends to zero, positive from the right and negative from the left). So $x=1$ is a vertical asymptote, and the discontinuity is of the second kind.

At infinity the degree of the numerator exceeds that of the denominator, so the limit is infinite: no horizontal asymptote, and it is worth looking for the oblique one.

The oblique asymptote is $y=x+1$, on both sides.

The check with polynomial division is immediate:

The quotient is $x+1$, that is the asymptote, and the remainder $\frac{2}{x-1}$ tends to zero: everything works out in ten seconds.

The five mistakes that cost the most marks

1. Writing $\frac{0}{0}$ as the result. It is not a number, it is the observation that the work has yet to begin. A limit cannot “be equal to” an indeterminate form.

2. Writing $\frac{k}{0}=\infty$ without the sign. You need to know whether the denominator tends to zero from above or from below, and so the two one-sided limits must be calculated. It is the step that distinguishes those who have understood vertical asymptotes from those who guess them.

3. Forgetting the absolute value when $x\to -\infty$. We have $\sqrt{x^2}=|x|$, which for negative $x$ is $-x$. It is the classic mistake in the oblique asymptote of an irrational function: on the right and on the left two different lines come out, and whoever does not put in the absolute value finds only one.

4. Looking for the oblique asymptote when there is already a horizontal one. They exclude each other on the same side. If you have found $y=2$ for $x\to +\infty$, the oblique one on the right does not exist and looking for it is wasted time.

5. Applying a standard limit without adjusting the argument. In $\frac{\sin 3x}{x}$ the denominator must become $3x$, and the factor $3$ must be brought out. It holds for all the standard limits, without exception.

The two weeks before the test

1. Write the five standard limits on a blank sheet every day for a week. Two minutes. Whatever does not come out is the day’s priority.
2. Do ten limits writing only the indeterminate form and the technique, without solving them: $\frac{\infty }{\infty }$ → factoring out, rational $\frac{0}{0}$ → factorising, roots → rationalising. It is the exercise that saves the most time in the test.
3. Do five complete studies of asymptotes, with domain, one-sided limits and the oblique one. It is the exercise worth the most and the one almost nobody practises enough.
4. Add three piecewise-defined functions with a parameter to determine for continuity: it is the most likely question among the “difficult” ones.

When limits are sorted, the next step in the syllabus is derivatives, and from there the complete function study — where everything you have learned here comes back, but inside a single exercise.

If you want to see the same tools inside function study, in sequence: part 1: domain and symmetries, part 2: intercepts and sign study; limits and asymptotes, with all the cases, are in part 3: limits and asymptotes, and afterwards come monotonicity, maxima and minima, concavity and inflection points and the graph with worked examples.

How we work on this in lessons

The first lesson on limits I spend almost entirely on recognition: we look at twenty limits and for each we say only the indeterminate form and the technique, without doing a calculation. In an hour it covers what solving them would take four lessons, and in the test the difference shows in the time, which is the really scarce variable.

On asymptotes I always work with the graph alongside: every limit calculated has to be translated into a piece of the drawing, otherwise they remain four unconnected sums. Anyone who can mentally draw what they are calculating does not get the signs of the one-sided limits wrong. Lessons are online and one-to-one: you will find up-to-date rates and packages on the pricing page, useful above all if Year 5 has just begun and it is worth arriving prepared through to function study with a package of several hours.

The quickest way to see where you stand is to try the simulated test on limits: fifteen questions in fifty minutes, with a breakdown of weak points by topic at the end. The correct answer to every question is free and needs no registration; the guided step-by-step solution is unlocked with the first lesson. If after the test you see that the calculation holds but the asymptotes do not (or vice versa), write to me for a free introductory call: we look at the syllabus of your test and I tell you honestly how many lessons you need. The full picture of how I set up the path is on the page for maths tutoring.