Ask for example what we obtain when adding a number to infinity. The common perception is that infinity plus any number is still infinity. That in turn would imply that all integers are equal, for example, and our whole number system would collapse. We could argue that it's 1, or 2, and again we have a contradiction since 1 does not equal 2. So here is a slightly more subtle approach.
Division is a continuous process.
Suppose b and c are both non-zero. Then, in a sense that can be made precise. There are many ways in which we can choose a and b and let them become smaller. It's a common strategy in teaching to simplify concepts when they are first encountered. In other words, it's common for your teacher to lie to you.