Incorrect.

The mathematics taught to school children is that zero times any number is zero.

This

can be proved from other properties of addition and multiplication.

For example, because $$\forall x\; x + 0 = x$$ and $$\forall x\; x \times 1= x$$ and $$\forall x,y,z \; z \times ( x + y ) = z \times x \, + \, z \times y$$ and $$\forall x,y,z \; x = y \leftrightarrow z + x = z + y$$ it follows that $$ x + 0 = x = x \times 1 = x \times ( 1 + 0 ) = x \times 1 + x \times 0 = x + x \times 0 \rightarrow 0 = x\times 0$$

This property of zero is why it has no finite representation in logarithms.

Multiplication by any number except zero can be "undone" by division. But since multiplication by zero throws away all information about the original number, there is nothing for arithmetic to operate on to recover the unique original number. Because $$5 \times 0 = 7 \times 0$$ does not allow you to assert $$5 = 7$$.

The topic of limits of expressions as parameters approach some value is not taught to "poor unexpecting children" but to young adults, usually in an introduction to differential calculus and illustrating the utility of L'Hôpital's Rule which is all about avoiding trying to evaluate expressions that look like they might approach a form like $$0 \times \infty$$ or $$\frac{\infty}{\infty}$$ or $$0^0$$ or $$\frac{0}{0}$$ and and using alternative expressions which have the same limiting value (which is often

*not* one) but better behavior. But zero is not an expression that approaches a value as some parameter is changed, so your invocation of analysis is unwarranted in this case and contrary to good instruction.