The chapter on limits introduces no new functions: the indeterminate forms of the first two weeks are almost all exponentials, logarithms and goniometric ratios already seen. Whoever arrives at without having in mind how the sine behaves near zero learns the result by heart and no longer recognises it as soon as the argument changes — and the function study in January, which is full of them, becomes a jumble of cases to learn one by one. In physics you start from electromagnetic induction, which is all one sentence: when the flux of the magnetic field changes, an electromotive force is born — without the flux from Year 4 that sentence can’t even be read.
The syllabus, block by block
Logarithms and exponentials. Half the indeterminate forms of the first month contain one. A logarithm is not a mysterious object: it is the answer to “to what exponent must I raise the base to get this number?”.
The latter is the property that brings the unknown down from the exponent — the reason logarithms exist. But the logarithm of a sum does not split: , while .
Goniometry for notable limits. They arrive in the second week and are used a hundred times a year. The point is not to memorise them: it is to recognise them when the argument is not exactly .
With the argument doubled you need a compensating factor: , double what the eye suggests.
Magnetic field and flux. The Faraday-Neumann law says only one thing: the induced electromotive force is the opposite of the rate at which the flux of the magnetic field changes.
The whole new chapter on induction is, inside, the derivative of this second formula.
The mistakes that cost the most
The logarithm of a sum is not the sum of the logarithms: the additive property holds for the product, never for addition inside the argument.
The notable goniometric limit is not applied by eye: you must check that under the fraction there is exactly the same argument as the sine, otherwise a correction factor is missing (or in excess).
The flux angle is measured from the normal to the surface, not from the plane: if the field “grazes” parallel to the loop, it forms with the normal and the flux is zero — it is the case from which half the chapter on induction starts, because the loop is rotated precisely to take the flux from zero to the maximum. Whoever measures the angle from the plane gets the wrong cosine in every single exercise with a rotating loop.
To review logarithms and exponentials there is the Year 4 practice test on exponentials and logarithms, for the notable values the goniometry quiz, and for the magnetic field the one on magnetostatics; on the limits and induction that open the year you’ll find the guides to limits, continuity and asymptotes and to electromagnetic induction.
The Italian version of this post comes with a downloadable document that covers all nine topics in full: every formula, one worked example per topic, the typical mistake where needed and twenty-seven exercises with the complete step-by-step solution.
If you would rather have guided revision than go it alone, maths and physics tutoring start exactly from here — write to me and we’ll see where to concentrate the first lessons.
