Infinite limit from the right calculator for rational functions

Enter numerator and denominator coefficients in descending powers, then provide the candidate x-value. The calculator checks that the denominator really vanishes there, accounts for any common factors by comparing zero multiplicities, and reports either positive infinity or negative infinity with the orders that justify the result. It is designed for dependable one-sided limit checks without requiring a factored expression.

Enter the rational function and approach value

Describe the numerator and denominator as coefficient lists ordered from the highest power of x to the constant term. For example, x squared plus 3x minus 4 is entered as [1, 3, -4], and x minus 1 is entered as [1, -1]. Include a zero whenever a power is missing: x cubed minus 4x must be [1, 0, -4, 0]. This explicit form avoids ambiguity about parentheses, implicit multiplication, and exponent notation. The asymptote field is the real number approached from the right, not an expression or an interval. The calculator accepts polynomials through degree twelve and requires a nonconstant denominator. It also verifies the mathematical premise instead of assuming the supplied value is valid. If the denominator is nonzero at that point, the request is rejected because no vertical asymptote exists there. Likewise, if cancellation removes every denominator factor at the point, the discontinuity is a hole or a finite limit rather than the requested infinite limit. Coefficients may be integers or finite decimals, but their positions must preserve every power in descending order.

How the right-hand sign is determined

Near the supplied value a, each polynomial can be described by its first nonzero local term. If the numerator begins with C times (x minus a) to the power m and the denominator begins with D times (x minus a) to the power n, cancellation leaves a factor proportional to C divided by D times (x minus a) to the power m minus n. A vertical asymptote remains precisely when n is greater than m. For a right-hand limit, x minus a is positive, even when its exponent is odd, so the direction of infinity is determined by the sign of C divided by D. A positive ratio produces positive infinity; a negative ratio produces negative infinity. The calculator finds m and n by repeated deterministic differentiation at a, which is equivalent to measuring factor multiplicity without asking you to factor either polynomial. The returned zero orders expose this reasoning, while remaining_denominator_order shows how many denominator factors survive cancellation. This method also handles repeated roots correctly, an important case where simply checking values immediately beside the asymptote can be numerically misleading.

Interpret the result and check your work

The limit field is the primary answer and contains either +infinity or -infinity. The sign field gives the same conclusion in words, approach confirms that the calculation used values from the right, and asymptote repeats the tested x-coordinate. Compare numerator_zero_order with denominator_zero_order to see whether cancellation occurred. A numerator order of zero means the numerator stays nonzero at the asymptote. A positive numerator order means the numerator and denominator share at least one factor there, but the denominator order must still be larger for a vertical asymptote to survive. The explanation field turns the computed sign into a short sentence suitable for a worksheet check or application display. Remember that a left-hand limit can differ when the surviving denominator order is odd; this capability intentionally answers only the right-hand question. It does not find all asymptotes automatically, draw a graph, or simplify the full rational expression. Use the structured result to verify a hand calculation, feed a tutoring workflow, or test generated calculus problems consistently. Identical inputs always produce identical results, with no network request or stored state.