Limit Calculator

How this limit calculator works

A limit describes the value a function approaches as x gets close to a point, even when the function itself is undefined there. This calculator evaluates lim(x→a) f(x) numerically: it samples the function at points closing in on a from both sides (a ± 0.1, ± 0.01, ± 0.001, ± 0.0001) and reports the value both sides converge to. Because it approaches the point rather than just plugging it in, it correctly handles removable discontinuities like (x²−1)/(x−1) at x = 1, where direct substitution gives the meaningless 0/0.

You can also request left-hand (x→a−) or right-hand (x→a+) limits separately, and limits as x approaches infinity. The approach table is shown with every answer so you can see the convergence yourself.

Entering your function

• Use ^ for powers: x^2, x^3
• Use * for multiplication, or just write 2x, 2(x+1), (x+1)(x-1)
• Parentheses group freely: (x^2 – 1)/(x – 1)
• Functions: sin(x), cos(x), tan(x), asin(x), acos(x), atan(x), ln(x), log(x), sqrt(x), abs(x), exp(x)
• Constants: pi, e
• For limits at infinity, enter inf or -inf as the point

Worked examples

Example 1: lim(x→1) (x²−1)/(x−1). Direct substitution gives 0/0, an indeterminate form. Factoring: (x−1)(x+1)/(x−1) = x+1, so the limit is 2. The calculator approaches from both sides (1.0001 gives 2.0001, 0.9999 gives 1.9999) and converges to 2. A naive tool that only substitutes x = 1 wrongly reports that this limit does not exist.

Example 2: lim(x→0) sin(x)/x. This is the classic trigonometric limit, equal to 1 (provable with the squeeze theorem). Sampling: sin(0.001)/0.001 = 0.99999983, converging to 1.

Example 3: lim(x→0) 1/x. From the right the values blow up to +infinity; from the left they blow up to −infinity. The two-sided limit does not exist, but the right-hand limit is +infinity and the left-hand limit is −infinity. Select the direction in the tool to see each side.

Example 4: lim(x→∞) (3x²+1)/(2x²). The highest-degree terms dominate: 3x²/2x² = 3/2. Sampling at x = 10⁴, 10⁶, 10⁸ converges to 1.5.

Techniques behind the answers

Direct substitution: when a function is continuous at the point, just plug in: lim(x→3)(x²+2x) = 15. Factoring: cancel common factors in 0/0 rational forms, as in Example 1. Rationalizing: multiply by the conjugate for square-root forms, e.g. lim(x→9)(√x−3)/(x−9) = 1/6. L’Hôpital’s rule: for stubborn 0/0 or ∞/∞ forms, differentiate top and bottom: lim(x→0) sin(x)/x becomes cos(0)/1 = 1. Standard limits: memorize sin(x)/x → 1 and (1−cos x)/x² → 1/2 as x→0; they unlock most trigonometric limits.

When a limit does not exist

A two-sided limit fails to exist in three situations: the left and right approaches disagree (a jump, or opposite infinities as in 1/x at 0), the function oscillates forever (sin(1/x) near 0), or it is unbounded on both sides with no agreement. The calculator distinguishes these: it reports +infinity or −infinity when values clearly blow up, and “does not exist” when the sides disagree or never settle, always showing the one-sided values so you can see why.

Limitations

This tool evaluates limits numerically, not symbolically: it is excellent for homework checking and intuition, but it is not a computer algebra system. Extremely delicate cancellations, cube roots of negative numbers, and scientific notation (write 0.00001 instead of 1e-5) are not supported. Always sanity-check surprising answers against the approach table shown with the result.

Graph, variable choice, precision, and Taylor series

A number is not the whole answer in calculus. This section adds what the main calculator cannot show: a graph of the function near the limit point with the limit marked, a variable selector (use t, h, or n instead of x when your class does), a decimal precision control, and a Taylor series expansion that reveals why the limit is what it is. It reads the function and point from the main calculator above.

Worked example. For sin(x)/x as x approaches 0, the graph shows the curve flattening toward the red dashed line at 1 from both sides. The degree-4 Taylor expansion at 0 is 1 – 0.166667x^2 + 0.00833333x^4 (the odd terms vanish), and the constant term, 1, is exactly the limit. That is the deep reason the limit is 1: near 0, sin(x)/x behaves like its Taylor polynomial, and the polynomial’s value at 0 is 1. Checking the polynomial at x = 0.1 gives 0.998334, matching sin(0.1)/0.1 to six digits.

Derivatives are computed numerically with central differences, so the coefficients carry tiny rounding noise in far digits. The graph samples 240 points and skips undefined ones, which is why vertical asymptotes show as gaps rather than spikes.

Frequently asked questions

What is a limit in calculus?

A limit is the value a function approaches as its input gets arbitrarily close to some point. It describes behavior near the point, not necessarily at it, which is why limits can exist even where a function is undefined.

How do I enter a function in the calculator?

Type it with x as the variable: use ^ for powers, * for multiplication (or write 2x), and parentheses to group. Supported functions include sin(x), cos(x), tan(x), ln(x), log(x), sqrt(x), abs(x), and exp(x), plus the constants pi and e.

Can it solve (x^2 – 1)/(x – 1) as x approaches 1?

Yes. Direct substitution gives 0/0, but approaching from both sides converges to 2, which matches the factored form x + 1. The calculator reports 2.

What are one-sided limits?

The left-hand limit (x to a-) uses only values below a; the right-hand limit (x to a+) uses only values above a. When they disagree, as with 1/x at 0, the two-sided limit does not exist.

What does it mean when a limit does not exist?

Either the left and right approaches disagree, the function oscillates without settling (like sin(1/x) near 0), or it grows without bound in an inconsistent way. The tool shows both one-sided values so you can see which case applies.

Can I compute limits at infinity?

Yes. Enter inf or -inf as the point. For example, (3x^2+1)/(2x^2) as x approaches infinity evaluates to 3/2.

Why did my trigonometric limit fail on another calculator?

Many simple tools substitute x directly into the expression and cannot evaluate functions like sin(x)/x at x = 0, reporting an error instead of the correct limit of 1. This calculator approaches the point numerically and supports trigonometric, logarithmic, and exponential functions.