The tidiness of elementary mathematics

We all expect a certain tidiness to mathematics, and for many this provides a gratifying aesthetic pleasure. However, there are some wrinkles. One of these arises in the integral of $x^k$. Calculus provides us with the formula $\int x^k\mathrm{d}x = \frac{x^{k+1}}{k+1}.$ However, this equation is only correct for $k\neq -1$. If we try to take $k=-1$ the right hand side is meaningless because we have a zero on the denominator of the fraction, i.e. $\frac{1}{0}$. But in this case a separate argument gives the answer $\int x^{-1}\mathrm{d}x = \int \frac{1}{x}\mathrm{d}x = \ln(x).$

Our expectation is that these two formulae should be reconciled. Indeed, if we let $k$ approach $-1$ in the first we should end up with the second, but that fails to happen. For each $x$, $\lim_{k\rightarrow -1} \frac{x^{k+1}}{k+1}$ is undefined.

This vingette explores this issue.

Applets

In this applet move the slider so that $k \rightarrow -1$.

In this applet move the slider so that $n \rightarrow \infty$.