Rational Function Limit

Finding a rational limit

First cancel factors common to the numerator and denominator. If the reduced denominator is nonzero at the limit point, substitute that point into the reduced expression. Polynomial coefficients are handled as fractions. The tool displays both a numeric value and a fraction.

Example

Factor x²−1 as (x−1)(x+1). Cancelling x−1 in (x²−1)/(x−1) leaves x+1. As x approaches 1 the limit is 2, even though the original expression is undefined at 1.

Scope

This is a calculator for finite polynomial and rational limits at finite points. It does not solve general symbolic limits, radical or trigonometric expressions, logarithms or limits at infinity. A zero reduced denominator prevents a finite result.

Separating indeterminacy from the result

A zero numerator and denominator after direct substitution do not mean the answer is zero. They mean substitution alone is insufficient. A common factor is cancelled for neighbouring values outside the approach point. Since a limit describes that neighbourhood, an undefined function value at the point can coexist with a finite limit.

For example, let x approach 3 for the expression (x²−9)/(x−3). Factor the numerator as (x−3)(x+3). Cancelling the common factor x−3 leaves x+3. Substituting 3 now gives 6. The original fraction still has a zero denominator at x=3. A limit of 6 does not automatically define the original function at that point.

Use parentheses around the entire numerator and denominator. Explicit multiplication reduces input mistakes with several terms. Cancellation applies to common factors, not individual terms of a sum.

If the reduced denominator is still zero, failure to return a finite result is not a classification of every possible limit. Infinite behaviour from the left and right needs separate analysis. A fraction output is an algebraic value for the supplied coefficients, not unlimited physical precision for measured coefficients. Choose an appropriate analytical or symbolic method for other function types.