Formula Used
A piecewise function is tested around a chosen point c.
f(x) =
left branch when x < a,
defined value when x = a,
and right branch when x > a.
The left-hand limit is lim x→c⁻ f(x). The right-hand limit is lim x→c⁺ f(x).
The two-sided limit exists when both one-sided limits exist and are equal within the chosen tolerance.
Continuity is checked by comparing the limit with f(c).
How to Use This Calculator
Enter the expression used before the breakpoint.
Enter the exact function value at the breakpoint, if defined.
Enter the expression used after the breakpoint.
Set the breakpoint and the limit point.
Choose left-hand, right-hand, or two-sided behavior.
Adjust tolerance and samples for stricter checking.
Press calculate. Review the result and sample table.
Use the CSV or PDF buttons to save the report.
Understanding Limits of Piecewise Functions
Piecewise functions are common in algebra, calculus, physics, economics, and engineering. They use different rules on different intervals. A limit question asks what value the function approaches near a point. It does not always ask for the value at that point. This difference is important.
Why One-Sided Limits Matter
A piecewise function can behave differently from the left and from the right. The left-hand limit studies values just less than the target point. The right-hand limit studies values just greater than the target point. If these estimates do not agree, the ordinary two-sided limit does not exist.
Breakpoint Behavior
The most important point is often the breakpoint. This is where the rule changes. A function may have a jump at this point. It may also have a hole. Sometimes the branches meet perfectly, but the defined value is different. In that case, the limit may exist, while continuity fails.
Numerical Checking
This calculator samples points near the selected limit point. It starts with a step size. Then it halves that step many times. The values should settle near one number when a finite limit exists. A smaller tolerance demands closer agreement. More samples can improve confidence.
Continuity Testing
A function is continuous at a point when three things are true. The function value is defined. The two-sided limit exists. The limit equals the function value. If any condition fails, the function is not continuous there.
Expression Support
You can enter powers, fractions, parentheses, constants, and common functions. Supported functions include sin, cos, tan, sqrt, abs, log, ln, and exp. Use x as the variable. Use pi and e for constants. Choose radians or degrees for trigonometric work.
Best Practice
Always compare the branch formulas first. Then check the sampled table. If values change sharply, reduce the first step. If errors appear, inspect domains. Square roots need valid inputs. Logarithms need positive inputs. Fractions cannot divide by zero. These checks help make the final limit decision clearer.
FAQs
What is a piecewise limit?
It is the value a piecewise function approaches near a point. The left and right branches may give different values, so both sides must be checked.
Does the function value affect the limit?
The function value does not decide the limit. A limit depends on nearby values. However, the function value matters when checking continuity.
When does a two-sided limit exist?
It exists when the left-hand limit and right-hand limit both exist and match. This calculator compares them using your selected tolerance.
What does tolerance mean?
Tolerance is the allowed difference between close estimates. A smaller tolerance is stricter. Very small tolerance may require more accurate inputs.
Can I use trigonometric functions?
Yes. You can use sin, cos, tan, sec, csc, cot, asin, acos, and atan. Choose radians or degrees before calculating.
Why do I see an error in the table?
An error usually means an expression is undefined at that sample point. Common causes include division by zero, invalid roots, or invalid logarithms.
Can the limit exist when continuity fails?
Yes. The left and right limits may match, while the function value is missing or different. That creates a removable discontinuity.
What variable should I use?
Use x as the variable. For example, enter x^2, sin(x), abs(x), or (x^2 - 4) / (x - 2).