To understand one-sided limits, let's start with an intuitive idea. In a regular two-sided limit, we approach a point from both directions - both arrows matter. But in a one-sided limit, we only care about approaching from one specific direction. Here we focus on approaching from the left side only.
Now let's look at the formal definition. The left-hand limit of f(x) as x approaches a from the left equals L if for every epsilon greater than zero, there exists a delta greater than zero such that when x is within delta units to the left of a, then f(x) is within epsilon units of L. The right-hand limit has a similar definition but approaches from the right side.
Let's compare one-sided and two-sided limits using a piecewise function. This function equals x plus 1 for x less than 2, and 5 minus x for x greater than or equal to 2. The left-hand limit as x approaches 2 is 3, and the right-hand limit is also 3. However, the two-sided limit does not exist because the function has a jump discontinuity at x equals 2.