Limits are an important idea in calculus, especially when looking at functions that don’t behave normally. Sometimes, limits just don't exist for a function at certain points where it is broken or has a gap. Here are three reasons why that can happen:

1. Jump Discontinuity:

   • A jump discontinuity is when a function jumps from one value to another. For example, let’s use this function, $f(x)$: $$f(x) = \begin{cases} 2 & \text{if } x < 1 \\ 5 & \text{if } x \geq 1 \end{cases}$$
   • If you look at what happens as $x$ gets close to 1 from the left side, you see that $f(x)$ is 2. If you look from the right side, it’s 5. Since these two values are different, the limit at $x = 1$ does not exist.

2. Infinite Discontinuity:

   • In this case, the function keeps growing bigger and bigger (or goes negative) as it gets near a certain point. For example, take the function $g(x) = \frac{1}{x}$. It has an infinite discontinuity at $x = 0$. As $x$ gets close to 0, $g(x)$ moves toward infinity, which means: $$\lim_{x \to 0} g(x) \text{ is undefined.}$$

3. Oscillatory Discontinuity:

   • Some functions bounce around lots of different values near a discontinuity without settling on one. A well-known example is $h(x) = \sin\left(\frac{1}{x}\right)$ as $x$ approaches 0. It jumps between -1 and 1, so: $$\lim_{x \to 0} h(x) \text{ does not exist.}$$

In summary, we see that limits might not exist when the left and right values are different, when a function grows really big, or when it oscillates around without settling down. This shows us why it’s important to understand continuity and limits in math.