Direct substitution
Continuous functions can usually be evaluated by substituting the approach point, provided the required side remains in the real domain.
Limit Calculator with Steps
Evaluate two-sided, left-hand, right-hand, and infinite limits. See direct substitution, indeterminate forms, L’Hôpital applications, leading-term analysis, one-sided comparisons, and a graph of the approach—all calculated locally.
Enter a limit
Use ^ for powers and parentheses for named functions such as sin(x), exp(x), ln(x), sqrt(x), and abs(x).
Try an example
Keyboard shortcut: Ctrl/⌘ + Enter.
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Limit
Entered limit
Enter a function to begin.
Result
The limit will appear here.
Left-hand limit—
Right-hand limit—
Step-by-step limit analysis
Limit steps will appear here.
Approach graph
Calculate a limit to graph the function near its approach point.
Calculator methodology and review
- Developed and reviewed by
- Starlight Robotics calculator team
- Last reviewed
- Symbolic methods
- Direct substitution, repeated L’Hôpital differentiation, polynomial leading terms, and rational degree comparison.
- Numerical method
- Independent left/right or end-behavior sequences with convergence, signed-growth, real-domain, and oscillation checks.
- Input limits
- One selected variable, 300 characters, finite points or ±∞, real-valued approach analysis.
- Privacy
- The expression and results remain in your browser; no solving API is called.
How to use the limit calculator
- Enter a function such as
sin(x)/xor(x²−1)/(x−1). - Choose the variable and enter a finite point,
inf, or-inf. - For finite points, choose two-sided, from the left, or from the right.
- Select Calculate Limit and review the method, one-sided behavior, steps and graph.
How limits are evaluated
L’Hôpital’s rule
Supported quotients producing 0/0 or ∞/∞ differentiate numerator and denominator, repeating when necessary.
One-sided comparison
A two-sided limit exists only when the left-hand and right-hand behaviors agree. Different finite values or opposite infinities produce DNE.
End behavior
Polynomial and rational limits use leading degrees and coefficients. Other supported expressions use extended evaluation and convergence sampling.
Worked limit examples
Removable discontinuity
lim x→1 (x²−1)/(x−1)
Substitution gives 0/0. L’Hôpital gives 2x/1, so the limit is 2.
Standard trigonometric limit
lim x→0 sin(x)/x
The 0/0 form becomes cos(x) after L’Hôpital, giving 1.
Unequal one-sided limits
lim x→0 1/x
From the left the function approaches −∞; from the right it approaches +∞. The two-sided limit does not exist.
Rational end behavior
lim x→∞ (3x²+1)/(2x²−4)
Equal degrees mean the limit is the ratio of leading coefficients: 3/2.
Limit calculator FAQs
When does a two-sided limit not exist?
It does not exist when the left and right limits differ, one required side has no real-domain values, or the function oscillates without settling.
When is L’Hôpital’s rule used?
For supported quotients with a 0/0 or ∞/∞ indeterminate form. The engine may apply it repeatedly, up to six times.
Can a limit exist where the function is undefined?
Yes. Limits describe nearby behavior, so a removable hole can have a limit even without a function value at the point.
Why do left and right limits matter?
They expose jumps, vertical asymptotes and domain boundaries. A two-sided limit requires both one-sided limits to agree.
How are limits at infinity calculated?
Polynomial and rational functions use leading-term analysis. Other expressions use direct end behavior, L’Hôpital where valid, or numerical convergence checks.
Does my function leave the browser?
No. All parsing, differentiation, evaluation and graphing run locally.