Normally when handling limits we consider a property of an expression to be valid at a limit of a parameter if that property is continuously true as we approach a limit of that parameter. For example: we can't divide by zero but we can divide by $a$ for arbitrarily small values of $a$ and show that this is valid $\lim_\limits{a\rightarrow 0}$.
Consider this:
We know that $\int x^{a}\mathrm{dx}$ is polynomial for $a\neq -1$
So surely $\lim_\limits{a\rightarrow -1} \int x^{a}$ is polynomial?
But we know this limit is actually $\ln(x)+C$ which is not a polynomial.
So the polynomial-ness of the result of integrating is violated $at$ the limit, even though it holds as we approach the limit.
So in what sense is this limit well-behaved?
One approach: Because of the $+C$ in an indefinite integral, I may say if we want to that $$ \int x^a\;dx = \frac{x^{a+1}-1}{a+1}+C, \quad\text{for all } a \ne -1 , $$ and then we have the correct limit $$ \lim_{a\to -1}\frac{x^{a+1}-1}{a+1} = \ln x . $$