Understanding Piecewise Limits
A piecewise limit checks behavior near a chosen input. The function may use different formulas on each side. That makes one-sided testing important. A normal substitution can be wrong when the active rule changes at the break point.
Why One-Sided Values Matter
The left-hand limit uses values that are slightly less than the approach point. The right-hand limit uses values that are slightly greater. A two-sided limit exists only when both sides move toward the same number. The actual function value at the point may be different. It may also be undefined. That does not always affect the limit.
Practical Uses
This calculator helps students, teachers, and analysts review discontinuities. It is useful for step functions, absolute value rules, rational pieces, and models with thresholds. Many real formulas change after a price, speed, temperature, or time crosses a boundary. Piecewise limits show whether that change is smooth.
How the Tool Works
You enter each branch formula and its condition. The calculator tests sample x-values near the approach point. It selects the matching branch for the left and right sides. Then it compares the two estimated limits. It also shows the value from the branch that includes the exact point.
Reading the Result
If the left and right limits match within the selected precision, the two-sided limit is reported. If they do not match, the calculator marks the limit as not existing. For a one-sided request, only the chosen side is required. The sample table helps you confirm the path used.
Best Practice
Use simple algebraic expressions. Add every branch needed near the target point. Check inequalities carefully. Place specific equality rules before broad rules when needed. Increase decimal precision for close answers. Use the export buttons to save results for class notes, reports, or later checking.
Common Mistakes
A common mistake is testing only the branch that contains the point. Another mistake is mixing strict and inclusive inequalities. For example, x less than two and x greater than two leave no value at two. That may be correct, but it should be intentional. Also remember that numerical checks are estimates. They support algebra, but they do not replace exact proof in formal work. Use algebra when possible.