Education
Limit Calculator
Enter a polynomial or rational expression in x and the value x approaches. The calculator tries direct substitution first, then cancels a (x − a) factor when the form is 0/0, and falls back to a numeric left/right check.
Expression
A polynomial in x, or a ratio of two polynomials. Use parentheses around the numerator and denominator if you include a division.
e.g. (x^2 - 4) / (x - 2)
A finite number. Limits at infinity are not supported here. · e.g. 2
What this supports
- Polynomials and ratios of polynomials in x.
- Direct substitution when the denominator is nonzero.
- One round of (x − a) cancellation when the form is 0/0.
- A numeric left/right check as a fallback.
Trigonometric, exponential, and logarithmic limits are not supported here. For those, use the scientific calculator numerically or work through the standard limit laws by hand.
Limit
4
After cancellation
For derivatives that depend on a limit definition, see the derivative calculator. For Riemann-sum style limits, the integral calculator is closer in spirit.
Examples
lim (x² − 4) / (x − 2) as x → 2
= 4 (removable)
lim x² + 3x + 1 as x → 1
= 5 (substitution)
lim 1 / x as x → 0
left -∞, right +∞: does not exist
lim (x² − 9) / (x − 3) as x → 3
= 6 (removable)
How it works
For continuous functions the limit is just the function value. For rational expressions the calculator checks the denominator first; if it is zero, it tries to factor out the shared (x − a) term.
Substitution · lim f(x) as x → a = f(a)
Removable · lim (x − a) · g(x) / ((x − a) · h(x)) = g(a) / h(a)
Numeric check · evaluate at a − ε and a + ε with ε small
Related calculus calculators
- Derivative calculator for the limit-based slope of a function at a point.
- Integral calculator for definite integrals as the limit of Riemann sums.
- Scientific calculator for numeric checks at points close to the target.
- Log calculator for logarithms that appear in many limit identities.
- All education calculators.
Frequently asked questions
A limit asks what value f(x) approaches as x gets close to a target. If the left and right approaches agree, the limit exists and equals that value. Limits are the foundation of derivatives and integrals.
When the function is continuous at x = a. Polynomials are continuous everywhere; rational functions are continuous everywhere the denominator is nonzero. In those cases, lim f(x) as x → a is simply f(a).
A 0/0 form at x = a usually means both numerator and denominator share a factor of (x − a). After canceling, the resulting expression evaluates at a and gives the limit value. The graph has a hole at x = a but the limit still exists.
Polynomials in x like 2x³ − x + 5, and ratios of two polynomials like (x² − 4) / (x − 2). Trigonometric, exponential, logarithmic, and piecewise functions are not parsed here.
If the left and right numeric approaches disagree (a jump discontinuity or oscillation), the two-sided limit does not exist. The calculator shows both side values so you can confirm.
Not in this version. For lim as x → ∞, compare the degrees of numerator and denominator: equal degrees give the ratio of leading coefficients; larger denominator gives 0; larger numerator diverges.
Related calculators
Education
Final Grade Calculator
Calculate your final course grade from your current grade and your final exam score.
Education
Weighted Grade Calculator
Combine assignments, quizzes, and exams by weight to get a precise weighted course average.
Education
Final Exam Calculator
Find out exactly what you need to score on your final exam to hit your target course grade.