Limit Calculator Formula

Understand the math behind the limit calculator. Each variable explained with a worked example.

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Formulas Used

Limit Val

limit_val = a * pow(x_approach, 2) + b * x_approach + c

Left Approx

left_approx = a * pow(x_approach - 0.001, 2) + b * (x_approach - 0.001) + c

Right Approx

right_approx = a * pow(x_approach + 0.001, 2) + b * (x_approach + 0.001) + c

Variables

VariableDescriptionDefault
aCoefficient of x²1
bCoefficient of x-3
cConstant2
x_approachx approaches2

How It Works

Limits of Polynomial Functions

Direct Substitution

For polynomial functions, the limit as x approaches a value c is simply f(c). This is because polynomials are continuous everywhere.

lim(x→c) f(x) = f(c)

Verification

The left and right approach values should agree with the direct substitution result, confirming the limit exists.

Worked Example

Find lim(x→2) of x² - 3x + 2.

a = 1b = -3c = 2x_approach = 2
  1. 01f(2) = (2)² - 3(2) + 2
  2. 02= 4 - 6 + 2
  3. 03= 0
  4. 04Left: f(1.999) ≈ -0.001999
  5. 05Right: f(2.001) ≈ 0.002001

Frequently Asked Questions

What is a limit?

A limit describes the value that a function approaches as the input approaches a particular point. It is the foundation of calculus.

Can you always use direct substitution?

For polynomials, yes. For rational functions, you may get 0/0, which requires algebraic simplification, L'Hopital's rule, or other techniques.

What if the left and right limits differ?

If the left-hand limit and right-hand limit are different, the two-sided limit does not exist. This happens at jump discontinuities.

Learn More

Guide

Introduction to Derivatives - Complete Guide

Learn the fundamentals of derivatives in calculus. Covers the definition, power rule, product rule, chain rule, and practical applications with clear examples.

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