One-sided limits allow us to examine function behavior as we approach a point from specific directions. The left-hand limit uses the notation lim x approaches a from the left, while the right-hand limit uses lim x approaches a from the right. In this continuous function example, both one-sided limits exist and are equal at point a.
When the left-hand limit and right-hand limit approach different values, the two-sided limit does not exist. This creates a jump discontinuity at point a. The function jumps from one value to another, creating a gap that cannot be bridged. This fundamental difference in limiting behavior means the overall limit is undefined.
Let's examine a concrete piecewise function with a jump discontinuity. For x less than 2, f of x equals x plus 1. For x greater than or equal to 2, f of x equals x plus 3. The left-hand limit as x approaches 2 is 3, while the right-hand limit is 5. This creates a gap of 2 units, demonstrating unequal one-sided limits.
There are two main types of discontinuities. Removable discontinuities occur when both one-sided limits exist and are equal, but there may be a hole or different function value at that point. Jump discontinuities are non-removable because the one-sided limits are unequal, creating a permanent gap that cannot be fixed by redefining the function at a single point.
Advanced examples include the absolute value function divided by x at x equals zero. The left-hand limit equals negative one, while the right-hand limit equals positive one. This concept is crucial in calculus for analyzing function behavior, determining continuity, and modeling real-world phenomena where sudden changes or thresholds occur.