NewLife✨

📊 Limit Calculator

Enter the numerator and denominator coefficients of a rational function and the point x approaches, and this tool estimates the limit.

Numerator coefficient a (of x²)

Numerator coefficient b (of x¹)

Numerator coefficient c (constant)

Denominator coefficient d (of x²)

Denominator coefficient e (of x¹)

Denominator coefficient f (constant)

x approaches

What this calculator does

This tool estimates the limit of a rational function f(x) = (ax²+bx+c) / (dx²+ex+f) as x approaches a chosen point, including the classic 0/0 indeterminate case (like the default example, (x²−1)/(x−1) as x → 1, which simplifies to a finite limit of 2).

The method

If the denominator is not zero at that point, the calculator uses direct substitution, which gives the exact limit. If the denominator is zero, it evaluates the function extremely close to the point from both the left and right side and averages the two, the same numerical idea used to confirm a limit exists when a factor cancels out. If the two sides disagree or the function grows without bound, the limit is reported as not existing.

This numerical approach works well for removable discontinuities but cannot replace full algebraic factoring for every possible function.
Provided for general educational purposes.

Last reviewed August 2026