Division by zero is not impossible in mathematics. The evidence for why it is not impossible, lies in the fact that the result such division could still be represented by a mathematic symbols. Depending on what number the division was performed on, the result of division by zero is either argument-less infinite complex number or a singularity.
If this division was performed on complex number other than zero, division by zero is going to result in complex number with infinite magnitude and undefined argument. This is because the number zero itself have no meaningful argument in itself. This is however doesn’t suggest in any way that division by zero is impossible. The result of said division still representable by a mathematical symbol, known as infinity.
If division by zero was performed on zero as well, then we have a bigger issue to discuss but it is still not impossible. Division by zero over zero is going to result in something mathematicians called singularity. This means that all numbers in existence are legitimate result of division of zero over zero. This is why singularity often become a point where the predictability of mathematics itself breakdown.
To understand what it means by singularity, one only have to observe the implicit plot result of equation such as x=x. It could be algebraically derived that x=x is equivalent to x-x=0 which is equivalent to 0x=0. This means that the value of x satisfying the equation x=x is x=0/0. The implicit plot result of such equation is going to show that all numbers within the range of plot is eligible to become the result of said equation.
If it is argued that division by zero is still impossible because the result is not a single number, then it should be noted that a mathematics equation or a set of mathematics equations may accept more than one number or solution vectors as their answer. A quadratic equation could have two complex solutions. A polynomial equation of order-n for example could have at maximum n-number of solutions satisfying the requirement of said equation. A trigonometric equation like sin(x)=0 also have infinite number of solutions.
Since it is perfectly legal for mathematics equations to have either infinity or more than a single number as their solutions, division by zero could not be considered as impossible. We only have to look at the problem from different points of view to provide solution for this problem.
It is also interesting to note that division by zero over zero is actually the reason why mathematicians invented a mathematics techniques we call limit. By using the concept of limit, mathematicians approach the possible occurrences of singularity in mathematics functions, especially rational functions to find both the horizontal and vertical asymptotic lines. In other word, by using the concept of limit, mathematicians look into the context in which this division by zero case happen and provide a contextual answer for the problem at hand.
Additionally the concept of limit is not really something academic with no real world application. The concept of limit combined with integration and differential calculus, could be used by computational physicist to predict what is going happen when two celestial bodies collides.
Division by zero is not only possible, but has also been applied for both inventing new mathematics concepts and to solve problems in real world.