Why is dividing by zero undefined instead of infinity?

The fundamental reason: Division is defined as the inverse of multiplication. $a \div b = c$ means $b \times c = a$. So $1 \div 0 = c$ would mean $0 \times c = 1$. But $0 \times c = 0$ for ANY $c$. Theres no number $c$ satisfying $0 \times c = 1$. **What about the limit argument?** $$\lim_{x \to 0^+} \frac{1}{x} = +\infty, \quad \lim_{x \to 0^-} \frac{1}{x} = -\infty$$ The left and right limits dont agree, so even in the extended real line, the limit doesnt exist as a single value. **The Riemann sphere** does define $1/0 = \infty$ (a single point at infinity), but this requires adding a new element $\infty$ to the complex numbers and changing the definition of division. Its useful in complex analysis but not in standard arithmetic. **In short:** Division by zero is undefined because it would contradict the definition of division itself.