A limit does not exist in the following main scenarios:
- When the function exhibits wild oscillations near the point, bouncing between different values and not settling (e.g., infinite oscillation).
- When the left-hand limit and right-hand limit as the variable approaches the point are different values, so the two-sided limit cannot be assigned one value.
- When the function grows without bound, approaching positive or negative infinity, causing unbounded behavior at that point.
- When there is a gap or a hole in the function at the point (discontinuity or undefined value).
- When algebraic evaluation leads to division by zero where the numerator is not also zero, indicating no finite limit.
In essence, a limit exists only if the function approaches a single finite number from both sides as the input approaches the point. If any of the above conditions occur, the limit does not exist.
Examples include:
- Different limit values from the left and right.
- Vertical asymptotes where the function goes to infinity.
- Oscillations like those from sine(1/x) near zero.
- Discontinuities where a function has a break or hole.
This gives a comprehensive view of when limits do not exist.
