Limits explained well: from intuition to indeterminate forms

The limit is the first truly “adult” concept of calculus, and it is also the one that many students go through without really understanding it — they learn to make the “h”s disappear and to substitute numbers, but if you ask them what they have calculated they freeze. It is a pity, because the limit is the brick on which everything else rests: continuity, derivatives, integrals. If limits stay vague, everything that comes after stays vague.

In this article I start from the intuitive idea, go through the standard limits and the indeterminate forms that serve 90% of problems between the maturità (Italy’s school-leaving exam) and Calculus 1, and close with the mistakes I see most often when I prepare students.

If instead you are preparing the Year 5 test, with continuity, points of discontinuity and asymptotes to classify, the angle you need is that of limits, continuity and asymptotes: the Year 5 test.

The intuitive idea (before the formulas)

Forget epsilon and delta for a moment. The idea of a limit is simple: where is a function heading as $x$ gets close to a certain value?

Note the word: getting close. The limit does not tell you what the function equals at the point — it tells you what it tends towards near the point. They are two different things, and it is exactly here that all the subtlety lies.

Take the function

At $x=1$ the function does not exist (zero denominator). But if you approach 1 — with 0.9, 0.99, 0.999, and from the other side with 1.1, 1.01, 1.001 — the values of the function get closer and closer to 2. So:

even though at $x=1$ the function is worth nothing. The limit describes the tendency, not the value at the point. Keep this in mind: it is misunderstanding number one.

The formal definition (and why it exists)

The epsilon-delta definition is frightening, but it says exactly what I wrote above, only with the precision needed in mathematics:

In words: however narrow you want the tolerance $\epsilon$ around $L$, I can always find a neighbourhood $\delta$ around $x_0$ in which the function falls within that tolerance. It is a game for two: you fix how close you want to be to $L$, I tell you how close you must be to $x_0$ to manage it.

In high school this definition is often “endured” without being used. In Calculus 1 instead it really is needed, because all the theorems (uniqueness of the limit, comparison, continuity) are proved starting from it. My advice: do not learn it by heart as a formula, but read it until you feel it says the same thing as the intuitive idea. They are the same sentence, one in English and one in mathematical language.

Right and left limits

Sometimes the function behaves differently depending on which side you approach from. The right limit ($x\to x_0^{+}$) looks at values greater than $x_0$, the left ($x\to x_0^{-}$) at smaller ones.

The golden rule: the limit exists only if the right and left limits coincide. A classic example, the sign function or $\frac{1}{x}$ at zero: from the right it tends to $+\infty$, from the left to $-\infty$, so the limit at zero does not exist. It is not $\infty$, it is not $0$: it does not exist, full stop.

This is central in function study for vertical asymptotes: when the denominator vanishes, you must always check the two sides separately, because often the function “jumps” from $-\infty$ to $+\infty$.

Standard limits (learn them by heart, but understanding them)

There are some limits that recur everywhere and that are worth knowing by heart. The two fundamental ones:

And, on the exponential and logarithm side:

These limits are not only to be memorised: they have to be recognised inside larger expressions. 70% of the mistakes on standard limits come from not seeing that $\frac{\sin (3x)}{x}$ is “almost” the standard limit — you only need to adjust the factor:

The argument of the sine and the denominator must be the same thing: that is the whole trick.

Indeterminate forms

When you substitute and obtain one of these expressions, you have not finished: you have found an indeterminate form, that is an expression that by itself does not decide the result.

“Indeterminate” does not mean “impossible”: it means that you have to work the expression to resolve it. The main techniques:

1. Factorising and cancelling — for polynomial $\frac{0}{0}$ forms. Factor out, factorise, cancel the factor that vanishes. It is the case of the opening example:

   $$\frac{x^2-1}{x-1}=\frac{(x-1)(x+1)}{x-1}=x+1\to 2$$

2. Comparing infinities — for $\frac{\infty }{\infty }$ forms. Factor out the dominant term. In

   $$\frac{3x^2+x}{x^2-5}$$

   for $x\to \infty$ the highest degrees rule: the limit is $3$. General rule between polynomials: compare the degrees of numerator and denominator.

3. Standard limits — for forms with sine, cosine, exponentials, logarithms near zero.

4. Rationalisation — for $\infty -\infty$ with roots: you multiply and divide by the conjugate.

5. Hierarchy of infinities — when logarithms, powers and exponentials coexist:

   $$\ln x\ll x^a\ll e^x(x\to +\infty )$$

   The exponential always wins, the logarithm always loses.

In Calculus 1 two powerful tools are added: L’Hôpital’s theorem (for $\frac{0}{0}$ and $\frac{\infty }{\infty }$, differentiating above and below) and Taylor expansions, which are often the cleanest way to resolve limits that by hand would become a nightmare. But beware: L’Hôpital is handy and for that reason overused. It must be used only when you are really in a $\frac{0}{0}$ or $\frac{\infty }{\infty }$ form, and sometimes a simple factorisation is faster and less error-prone.

The commonest mistakes I see

After following quite a few students on this chapter, the mistakes repeat with an almost comic regularity. Knowing them in advance saves you marks.

1. Confusing “value of the function” with “limit”. The limit describes the tendency near the point, not the value at the point. A function can have limit 2 at $x_0$ and equal 7 at $x_0$ (that is exactly how points of discontinuity are defined).

2. Substituting at once and stopping at an indeterminate form. Getting $\frac{0}{0}$ is not the answer: it is the starting point. If you write "$=\frac{0}{0}$" and stop, the question is still all there.

3. Treating $\infty$ as a number. $\infty -\infty$ is not zero, $\frac{\infty }{\infty }$ is not one. Infinity is not a number: it is a behaviour. Expressions with $\infty$ are shorthand, not arithmetic operations.

4. Forgetting the two sides. For limits at points where the denominator vanishes, ALWAYS check right and left. Very many mistakes on vertical asymptotes come from here.

5. Applying a standard limit without adjusting the argument. $\frac{\sin (5x)}{x}$ is not 1: it is 5. The argument of the sine and the denominator must coincide before you can say “tends to 1”.

6. Using L’Hôpital across the board. It works only on $\frac{0}{0}$ and $\frac{\infty }{\infty }$. Applying it to a form that is not indeterminate gives wrong results, and examiners notice at once.

7. Skipping the written steps. In the Calculus 1 oral (and in the maturità too) how you arrive at the result counts. Writing every step — “we are in form $\frac{0}{0}$, I factor out, I cancel” — is not wasted time: it is the difference between an exercise that is “right by chance” and one that is mastered.

How we work in lessons

On limits I always start from intuition, not from the definition. First I make you see what happens — with a table of values getting closer, with the graph — and only afterwards do we introduce the formalism. When the idea is clear, the epsilon-delta definition stops being a monster and becomes simply “the precise way of saying something you already know”.

From there we build a repertoire of techniques: recognising the indeterminate form, choosing the right tool (factorising, comparison, standard limit, rationalisation), and — above all — understanding when to use which. I do a lot of exercises aloud together with the student: it is not enough for the sum to work out, they must be able to explain to me why they took that step. That is how you arrive prepared for an exercise you have never seen, which is then what happens to you in the exam.

We work online with a graphics tablet and a shared screen, writing the steps by hand as you would in the test. The 10-hour package typically covers a solid block of calculus — limits, continuity and the link to derivatives — while for a specific doubt or a pre-test review a single lesson is perfectly fine; up-to-date rates are on the pricing page.

If you want to see where you stand, I propose a free 30-minute introductory call: you show me a couple of exercises that do not work out for you, we work out together whether the problem is the concept or the calculation, and I tell you honestly how I would set up the work. No pushy selling.

For the full picture of how I work on maths there is the maths page; if instead you are preparing a university entrance test where limits weigh heavily, have a look at the admission tests. And when limits are solid, the natural next steps are derivatives and integrals: it is all the same story, told one chapter at a time.

If your course starts from limits of sequences, I have written separately about sequences in Calculus 1; for complex numbers, which many courses cover in the same weeks, there is complex numbers explained well.

Want to test yourself? Take the limits review quiz: 15 questions, with solutions explained step by step.