When students are learning about limits in math, there are some common misunderstandings. Here are a few things to keep in mind:

1. Limits are not the same as function values: A limit tells us what the function gets close to, not what it actually equals. For example, when we say $\lim_{x \to 0} \frac{\sin x}{x} = 1$, it means as $x$ gets closer to 0, the function gets closer to 1, even though $\frac{\sin(0)}{0}$ doesn’t have a value.

2. Limits can exist without a defined function: You can have a limit even if the function isn’t defined at that point. For example, in the case of $\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$, the limit equals $2$, even though $f(1)$ doesn’t have a value.

3. Limits don’t happen immediately: Limits show us how a function behaves as $x$ gets closer to a certain number. It’s not just about what happens at that number itself.