Limit Calculator

The most common mistake about limits is thinking they equal the function's value at that point. A limit is what the function approaches, not what it equals, and it can exist even when the function is undefined there. The limit calculator shows you exactly that separation by working through three methods in order: direct substitution, algebraic manipulation, and L'Hôpital's rule. You get the value plus the reasoning behind it, not just a number.

To enter a limit, pick a function type from the dropdown, polynomial, rational, trigonometric, exponential, logarithmic, radical, or piecewise, then type the expression. For the limit point, enter a number, inf, or -inf. Choose direction: two-sided, left-hand, or right-hand. The calculator handles one-sided limits and limits at infinity. Set decimal places from 2 to 8, and tick boxes to show the values table, graph, or solution steps. Untick 'numerical approximation' to force exact direct substitution where the function is continuous.

The limit solver follows a fixed hierarchy. First it tries direct substitution. If the function is defined and continuous at the limit point, meaning plugging in the x-value gives a real number, that number is the limit. No fancier technique is needed.

When Direct Substitution Fails

If direct substitution gives an indeterminate form like 0/0 or ∞/∞, the calculator moves to algebraic methods. It factors the numerator and denominator, cancels common factors, or rationalizes expressions involving radicals. For rational functions at infinity, it compares the degrees of the numerator and denominator, if the numerator degree is less, the limit is 0; if equal, the ratio of leading coefficients; if greater, the limit is infinite.

L'Hôpital's Rule

If the limit remains indeterminate after algebra, or if the expression is not easily factorable, the calculator applies L'Hôpital's rule. This differentiates the numerator and denominator separately and re-evaluates the limit. The rule requires the original form to be 0/0 or ∞/∞. It can be applied repeatedly if the result remains indeterminate. The calculator stops and reports the limit once a determinate value is reached, or declares the limit does not exist if it oscillates or diverges.

Numerical Approximation

When the exact method fails, for example with functions that oscillate or have essential discontinuities, the calculator falls back to numerical approximation. It samples the function at points approaching the limit from the chosen direction, looking for three consecutive values that agree within a small spread. It detects unbounded growth (∞ or -∞) and oscillation (DNE). This numerical method is not a proof but a reliable estimate, and the tool flags it as such in the solution steps.

After calculating, the results panel shows five pieces of information. The limit value is the main number, with left-hand and right-hand limits listed separately if the direction was two-sided. The function value at the point f(a) is displayed below, with 'Undefined' if the function is not defined there. The continuity field tells you whether the limit equals f(a), if yes, the function is continuous; if not, it names the discontinuity type: removable (hole), jump, or infinite.

Approaching Values Table

The table shows x-values getting closer to the limit point, with the corresponding f(x). For a two-sided limit, it lists values from both sides. This table lets you see the trend numerically, which is especially helpful when the graph is ambiguous.

Function Graph

The graph plots the function as a blue curve with a red dot at the limit point (a, limit value). It zooms in automatically near the point. Use it to check for jumps, holes, or vertical asymptotes that the numerical table might miss. The graph is particularly useful for piecewise functions where the algebraic expression changes at the limit point.

Evaluate lim_{x→0} (sin x)/x using the limit calculator.

Enter the function type as 'trigonometric' and the expression sin(x)/x. Set the limit point to 0 and direction to 'two-sided'. Click calculate.

The output shows a limit value of 1. The left-hand and right-hand limits are both 1.Continuity reads 'Removable Discontinuity', the limit exists but the function is not defined at the point. The approaching values table lists x = ±0.1, ±0.01, ±0.001 with f(x) getting closer to 1. The graph shows a curve approaching 1 from both sides with a hole at (0,1). The solution steps explain: direct substitution gives 0/0, so the calculator uses the limit of sin(x)/x = 1, which is a standard trigonometric limit taught in Stewart §3.3.

The limit solver with steps is the core of this tool. It does not just output a number, it shows the reasoning. The steps panel lists each technique tried, the form detected, and the algebra or rule applied. For L'Hôpital's rule, it shows the derivative of the numerator and denominator separately before re-evaluating. For factoring, it shows the factorisation and cancellation. This walkthrough helps you check your own work and understand why a particular method was chosen.

For piecewise functions, the calculator evaluates the appropriate piece for the chosen direction. If the direction is left-hand, it uses the expression for x < a; for right-hand, the expression for x > a. The optional 'at x = a' field lets you define f(a) separately for continuity checking.