Limits Calculator
Enter a function and the point approached by x. The calculator identifies the limit form, selects an available method and shows each step, comparing one-sided limits when necessary.
Set up the limit
More examples
Syntax and supported cases
Use ^ for powers, * for multiplication and parentheses to control the order of operations. Supported functions include abs, sqrt, ln, exp, sin, cos, tan, arctan, arcsin, log2, log10, sinh, cosh and tanh in the documented cases. Angles are in radians. Selected limits involving abs(f(x)) and radicals that are not differentiable at the target point are supported too.
The calculator handles rational functions of degree up to 8 and selected special forms with polynomial arguments. It also supports some variable-exponent powers, including x^x and x^(1/x). For composite quotients it recognizes direct substitution, 0/0 forms and poles; where justified, it uses Taylor and Maclaurin expansions through order 13. In certain sums of meromorphic terms it applies Laurent series and shows cancellation of negative powers. It handles selected sums, products and powers of bounded oscillations multiplied by vanishing factors, undamped oscillations and some oscillatory limits at infinity. Rigorous bounds are provided for certain oscillations damped by radicals, logarithms and exponentials, including bounded-away-from-zero oscillating denominators. It also studies products of exponentials with divergent exponents and meromorphic factors when dominance can be proved. Simple fractional coefficients are recognized from numerical coefficients. At infinity it compares sums of exponential, power and logarithmic terms in supported cases. Unsupported methods are reported explicitly.
Limit result
Step-by-step solution
Graphical interpretation
Illustrative graph: numerical samples show the trend but do not prove the limit. The curve may be interrupted where the function is undefined.
Near the point, oscillations become increasingly dense. The plotted samples may not capture all of them. Dashed curves indicate upper and lower bounds of the oscillating product.
Values near the point
These numerical approximations help interpret the result, but do not constitute a proof.
| x from the left | f(x) | x from the right | f(x) |
|---|
Function Reference line x = a Limit value Oscillation envelope Shaded area: oscillations not resolved by the plot
Available methods and limitations
The calculator handles polynomials, rational functions, standard limits, selected Taylor, Maclaurin and Laurent expansions, rationalization with radicals, and implemented cases of ∞−∞, 0·∞, 1^∞, 0^0 and ∞^0. It compares one-sided limits, identifies indeterminate forms and explains the methods used.
The explanations use algebraic identities, expansions, the squeeze theorem, sequences and symbolic rules where supported. Version 1.23 added sequence-based proofs of nonconvergence for oscillations with a common phase, including terms of the form 1/(x−a)^m, exact-value summaries of oscillatory behavior, and plots that mark unresolved oscillations and one-sided expansions involving absolute values and radicals. It can also certify the real domain of some inherently nonnegative expressions, such as 1−cos(x) and squares, without relying on the graph. Not every function composition is supported: l'Hôpital's rule is used only when its assumptions have been checked and the 0/0 form persists. Oscillation, pole analysis and behavior at infinity are limited to documented families; the numerical graph may miss singularities. A numerical plot is never a proof.