Polynomial Limits Calculator

Calculator Inputs

Numerator coefficients

Enter highest degree first. Example: 1, -3, 2

Denominator coefficients

Optional. Leave blank for a simple polynomial.

Limit type

Approach value a

Used for x → a.

Side behavior

Numerical check step

Smaller values inspect approach behavior more closely.

Decimal precision

Choose between 2 and 12 decimals.

Graph minimum x

Graph maximum x

Formula Used

1) Direct substitution

If the denominator is not zero at the approach value, substitute directly:

lim x→a [P(x)] = P(a)

lim x→a [P(x)/Q(x)] = P(a)/Q(a), when Q(a) ≠ 0

2) Factor cancellation for 0/0 forms

When both numerator and denominator become zero, remove common factors like (x − a)^k before substituting again.

P(x) = (x − a)^mR(x), Q(x) = (x − a)ⁿS(x)

Cancel the shared power min(m, n), then evaluate the reduced expression.

3) End behavior at infinity

For large positive or negative x, dominant terms decide the limit:

P(x)/Q(x) ≈ (aₙxⁿ)/(bₘxᵐ)

If n < m, the limit is 0.
If n = m, the limit is aₙ / bₘ.
If n > m, the magnitude grows without bound, with sign determined by coefficients and parity.

How to Use This Calculator

Enter numerator coefficients in descending powers of x.
Enter denominator coefficients if your expression is rational. Leave it blank for a plain polynomial.
Choose whether the limit is at a point, positive infinity, or negative infinity.
For finite limits, enter the approach value and choose two-sided, left-hand, or right-hand behavior.
Set precision and graph range to match your study or checking needs.
Press the calculate button to view the result, stepwise explanation, graph, and downloadable exports.

Use commas or spaces between coefficients. Example: 2, -5, 3 means 2x² − 5x + 3.

Example Data Table

Example	Expression	Approach	Expected result	Reason
1	(x² − 3x + 2)/(x − 2)	x → 2	1	Cancel (x − 2), then substitute.
2	3x² − x + 4	x → 2	14	Direct substitution works immediately.
3	(2x³ + 1)/(x² + 5)	x → ∞	∞	Numerator degree exceeds denominator degree.
4	(4x² − 1)/(2x² + 7)	x → -∞	2	Equal degrees use leading coefficient ratio.
5	(x + 1)/(x² + 1)	x → ∞	0	Denominator degree is higher.

FAQs

1) What coefficients format should I enter?

Enter coefficients from the highest power to the constant term. For example, 1, -3, 2 represents x² − 3x + 2.

2) Can this calculator handle rational expressions too?

Yes. Add denominator coefficients to evaluate limits of polynomial ratios, including removable discontinuities and end behavior.

3) What happens when substitution gives 0/0?

The calculator checks repeated factors at the approach value, cancels common powers when possible, and then evaluates the reduced expression.

4) Can it show one-sided limits?

Yes. Choose left-hand or right-hand mode for finite points when you want to inspect behavior near vertical asymptotes or mismatched side values.

5) How are limits at infinity determined?

The calculator compares the degrees and leading coefficients of the numerator and denominator. Dominant terms decide whether the result is zero, finite, or unbounded.

6) Why does the graph sometimes break?

Graphs break around undefined points so discontinuities and asymptotes remain visible. This prevents misleading straight lines across invalid values.

7) What does DNE mean?

DNE means the limit does not exist. This often happens when left-hand and right-hand behaviors disagree or the expression oscillates without settling.

8) Can I export the result?

Yes. After calculation, use the CSV and PDF buttons to save the computed result, method, steps, and numerical check table.