One-sided limits examine function behavior as we approach a point from either the left or right side. The left-hand limit uses the notation x approaches a from the negative side, while the right-hand limit uses x approaches a from the positive side. For the function f of x equals x squared at x equals 2, both one-sided limits equal 4, showing the function approaches the same value from both directions.
The fundamental theorem states that if the left-hand limit equals the right-hand limit, then the two-sided limit exists and equals that common value. This is both necessary and sufficient for limit existence. For example, with f of x equals 3x plus 1 at x equals 2, both one-sided limits equal 7, so the two-sided limit exists and equals 7. This equality condition is the key to determining when limits exist.
When one-sided limits are not equal, the two-sided limit does not exist. Consider this piecewise function: f of x equals x plus 1 for x less than 2, and 2x for x greater than or equal to 2. The left-hand limit as x approaches 2 is 3, while the right-hand limit is 4. Since these values are different, there is a jump discontinuity, and the two-sided limit does not exist. This demonstrates why the equality condition is essential.
The greatest integer function demonstrates another case where one-sided limits differ. This step function equals the largest integer less than or equal to x. At x equals 3, the left-hand limit is 2 since values just left of 3 give floor values of 2, while the right-hand limit is 3 since values just right of 3 give floor values of 3. The function jumps at every integer, making one-sided limits unequal and two-sided limits nonexistent at these points.
A removable discontinuity shows that equal one-sided limits guarantee limit existence regardless of the function's actual value at that point. Consider f of x equals x squared minus 4 over x minus 2, with f of 2 defined as 5. Simplifying gives x plus 2 for x not equal to 2. Both one-sided limits equal 4, so the limit exists and equals 4, even though the function value at x equals 2 is 5. This demonstrates the precise meaning of the equality theorem.