Making sense of $\lim_{\Delta\rightarrow\infty} \frac{\Delta}{\Delta}$

Question

We can construct the Dirac delta function by taking a rectangle with width $\Delta$ and height $\frac{1}{\Delta}$, which has area $1$ for all $\Delta\neq 0$, in the limiting case $\Delta\rightarrow 0$. Then it makes sense that the Dirac delta has integral $1$ under this construction since $\Delta$ never actually equals $0$ in the limit. What if we apply the same reasoning in the limiting case $\Delta\rightarrow\infty$? Then still we have integral $1$, but this resulting function demonstrates the same behavior as the function $f(x)\equiv 0$ whose integral is $0$. At first glance this seems paradoxical, but I can see where that's not so accurate to conclude. Still, I'm having trouble reasoning with rigor here. Does anyone have any insights?