Find a Limit Online and Check the Numerical Estimate

Enter the function and the approach value separately

For the limit of (x²−9)/(x−3) as x approaches 3, put (x^2-9)/(x-3) in the expression field and 3 in the approach field. Use the variable x and explicit parentheses around each compound numerator or denominator. The function is undefined at 3, but its limit depends on nearby values rather than direct evaluation at that point. Do not enter the complete sentence or a limit symbol in the function field; the form needs the expression it will evaluate numerically.

Verify the first result by factoring

For x≠3, x²−9=(x−3)(x+3), so the original quotient equals x+3. As x approaches 3, that simpler expression approaches 6. This establishes the exact limit 6 even though the original expression has a hole at 3. A browser estimate close to 6 supports the arithmetic check, while the factorization explains why the value is correct. Cancelling x−3 does not redefine the original function at 3 unless you explicitly create a continuous extension.

Try a second function with a known limiting value

For sin(x)/x as x approaches 0, enter sin(x)/x and approach value 0. The trigonometric expression uses radians. The standard limiting value is 1, and the nearby numerical outputs should support an estimate close to one. The undefined substitution 0/0 is not itself an answer. If the same formula were interpreted with the numerical argument in degrees, its limiting scale would differ, so the angle convention is part of the mathematical problem.

Compare behaviour on both sides

A finite two-sided limit requires the left and right behaviours to approach the same value. For abs(x)/x near 0, negative inputs give −1 and positive inputs give 1, so no single two-sided limit exists. For 1/x, the magnitudes grow with opposite signs as zero is approached. These are different reasons for failure, and neither should be replaced by an average of the last two numbers. A tool's sampled estimates help you inspect behaviour, but the expression determines whether the conclusion is valid.

Treat numerical sampling as evidence with limits

The calculator checks offsets from 0.1 down to 0.0000001 on each side of the approach value. Its heuristic comparison tolerance is 0.001 times one plus the magnitude of the compared value, rather than a requested error bound. An expression can vary sharply between those points, and an oscillatory function can accidentally look settled at a chosen sequence. The function sin(1/x) has no limit at zero because it continues oscillating, even if a short table appears suggestive. A numerical result is therefore most useful alongside continuity, algebra, known limits or another analytic argument. More displayed decimals do not turn sampled evidence into a proof.

Watch subtraction cancellation

The expression (sqrt(x+9)−3)/x near zero subtracts two nearly equal numbers. That can lose useful numerical digits before division by a tiny x. Rationalizing gives 1/(sqrt(x+9)+3) for x≠0, which has limit 1/6 and is numerically easier to evaluate near zero. If two equivalent expressions produce slightly different estimates, cancellation may explain the difference. Preserve the excluded point when simplifying and compare the expected limiting value, rather than assuming the expression with more digits is automatically better.

Respect domain boundaries and finite input scope

A real square-root or logarithm expression may be undefined on one side of an approach point. For sqrt(x) at zero, the right-hand behaviour is meaningful, while negative nearby real inputs are outside the domain. State whether the requested task is a one-sided limit or a limit relative to the function's domain. The local form has a finite numeric approach input and does not offer an infinity token or a separate direction selector, so it cannot replace a full symbolic treatment of every endpoint problem.

Record the estimate and the justification separately

A complete worked answer includes the original expression, the approach point, the numerical estimate if used, and the mathematical reason for the conclusion. For the first example, write that the quotient equals x+3 away from 3 and therefore tends to 6. For sin(x)/x, identify the radian convention and the standard limit. This reporting style makes a parser mistake, floating-point issue or unsupported problem type easier to distinguish from an error in the underlying mathematics.

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