Limit calculator with steps
A limit describes the value a function approaches near a point, even if the function is not defined at that point. This free limit calculator with steps handles documented real, single-variable forms. Choose a finite point or positive or negative infinity. At a finite point you can compare both sides or request only the left-hand or right-hand limit.
Limits and direct substitution answer different questions at a hole. For (x^2-1)/(x-1), the expression is undefined at x = 1, but its limit is 2. Your existing Derivative Calculator instead finds a rate of change. The Graphing Calculator can help explore the curve visually.
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For additional practice with holes and one-sided behavior, optional calculus workbooks (paid link) can provide exercises to solve before checking your own working. No purchase is required.
How to use this limit calculator online
Enter the expression and select x, y or t. Choose Finite point and enter the approach point, then choose Two-sided, From the left or From the right. For end behavior, select Positive infinity or Negative infinity; finite-point controls are then disabled. Select Calculate to see the result and the method used.
Editing an input clears the previous answer. Reset restores a blank function and the initial settings. Copy, Download and Print include the calculated steps. Write a complete quotient as (x^2-1)/(x-1), with parentheses around the numerator and denominator. A graph-paper notebook (paid link) is optional for keeping left and right working separate. The Polynomial Factoring Calculator can help with a rational expression before you inspect its nearby behavior.
Expression syntax and angle conventions
Use x, y or t as the single selected variable; numbers; pi and e; parentheses; and +, -, *, / or ^. Supported functions are sin, cos, tan, exp, ln, sqrt, abs, asin, acos and atan. Write function arguments in parentheses. log means the natural logarithm, just like ln. Trigonometric angles and inverse trigonometric values use radians.
Explicit multiplication, such as 2*x, and implicit multiplication, such as 2x or (x+1)(x-1), are accepted. Powers associate to the right: x^2^3 means x^(2^3). A leading minus has lower precedence than a power: -x^2 means -(x^2). Constant fields accept expressions such as 1/3, pi/2 and sqrt(2), but not the selected variable. Use exact rational points when checking a removable singularity.
How finite limits and limits at infinity are calculated
For supported finite expressions, the engine collects algebraic sums, products, integer powers and quotients into a numerator and denominator. It propagates local power expansions about the approach point through order 12. If the first nonzero numerator and denominator powers are m and n, the nearby ratio behaves like (A/B)*(x-a)^(m-n). Positive, zero and negative exponent differences give zero, a leading-coefficient ratio and unbounded magnitude, respectively.
The sign of an odd negative power differs on the left and right. abs of an analytic inner expression can also use side-dependent leading signs. Elementary compositions require their documented real local domains. Outermost ln and sqrt have a small additional boundary rule for a positive inner expression approaching zero. The method does not use a few sampled values as proof of a limit.
At infinity, this version supports rational functions of polynomials. A lower numerator degree gives zero, equal degrees give the ratio of leading coefficients, and a higher numerator degree gives signed unbounded behavior. Use the Fraction Calculator to check rational leading coefficients. L’Hôpital’s rule is not the algorithm displayed by this version.
Worked limit examples
Removable hole: For (x^2-1)/(x-1) as x approaches 1, write h = x-1. The numerator is 2h+h^2 and the denominator is h. Their lowest powers are both 1, so the limit is 2. This does not assign a value to the original expression at x = 1.
One-sided disagreement: For 1/x at 0, the left-hand limit is −∞ and the right-hand limit is +∞. The two-sided result is Does not exist (DNE). For 1/x^2 both sides increase without bound, so the two-sided infinite limit is +∞.
Standard trigonometric form: For sin(x)/x at 0, both leading local terms are x, giving a limit of 1 in radian mode. For (3x^2+2)/(2x^2-5) at positive infinity, equal degrees give 3/2. Optional scientific calculators (paid link) can help check ordinary fractions and function values; rounded numerical samples alone do not prove any of these limits.
Interpreting finite values, infinity and DNE
A fraction without ≈ is retained exactly within the supported arithmetic. The ≈ prefix identifies a rounded floating-point value. An infinite limit describes unbounded behavior, not a finite number to substitute back into the function. DNE means the requested two-sided comparison failed, rather than that every nearby value is undefined.
This tool’s two-sided mode requires real behavior on both sides. For sqrt(x) at 0, choose From the right to obtain 0; the left side has no real domain. A general mathematical limit relative to a restricted domain can use a different convention, so match the convention in your course. Optional graphing calculators (paid link) can help visualize a function near a point, but a finite graphing window cannot establish limiting behavior.
For a local polynomial approximation rather than a single limiting value, continue with the Taylor & Maclaurin Series Calculator. To study accumulated change over an interval, use the Integral Calculator.
Supported forms and calculation limits
Finite limits cover rational expressions and supported analytic elementary combinations whose collected numerator and denominator can be expanded locally. Standard inputs such as sin(x)/x, (exp(x)-1)/x and ln(1+x)/x at 0 are supported. Absolute-value side behavior and selected outer ln/sqrt boundary cases are included. This is not a general symbolic limit solver.
A denominator that vanishes through order 12 may exceed the method. Essential singularities, oscillatory singularities such as sin(1/x), general fractional-power singularities, piecewise inputs and multivariable limits are not solved. Infinity mode requires rational-coefficient polynomials; pi/e coefficients and trigonometric, logarithmic or exponential end behavior are outside that mode. Simplify an expression first when a function contains a removable singularity inside its argument. Zero powers in limit mode require a polynomial base, so domain exclusions are not silently discarded.
Unresolved expressions show an explanatory error; an error does not prove DNE. Near-zero floating-point coefficients are rejected where they make the result unreliable. Inputs allow at most 240 characters, 160 tokens and 24 nested levels. Points have magnitude at most 1,000,000. Integer-power and polynomial-size limits, exact-coefficient size limits and a worker timeout also apply.
Limit Calculator FAQs
Sources and methodology
These educational references explain the underlying mathematics. The independent implementation uses the supported methods and restrictions documented above and is not affiliated with the publisher.