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Limits & Continuity Explorer

Enter a function and a point to evaluate the limit from both sides. The tool builds a table of approaching values, graphs the function, and classifies any discontinuity.

Input

Limit Results

Left-hand limit
1
Right-hand limit
1
Two-sided limit
1
f(0)
DNE
Classification
Removable Discontinuity

Graph

Values Approaching x = 0

From the Left (x < 0)

xf(x)
-0.100000000.99833417
-0.0100000000.99998333
-0.00100000000.99999983
-0.000100000001.0000000
-0.0000100000001.0000000
-0.00000100000001.0000000

From the Right (x > 0)

xf(x)
0.100000000.99833417
0.0100000000.99998333
0.00100000000.99999983
0.000100000001.0000000
0.0000100000001.0000000
0.00000100000001.0000000

Step-by-Step

1. Evaluate the left-hand limit

lim⁡x→0−f(x)=1\lim_{x \to 0^-} f(x) = 1

2. Evaluate the right-hand limit

lim⁡x→0+f(x)=1\lim_{x \to 0^+} f(x) = 1

3. Two-sided limit

lim⁡x→0f(x)=1\lim_{x \to 0} f(x) = 1

4. Classification

Removable Discontinuity

Reference Guide

Left and Right Limits

The left-hand limit approaches from values less than a; the right-hand limit from values greater than a.

lim⁡x→a−f(x)andlim⁡x→a+f(x)\lim_{x \to a^-} f(x) \quad \text{and} \quad \lim_{x \to a^+} f(x)

Two-Sided Limit

The two-sided limit exists only when both one-sided limits exist and are equal.

lim⁡x→af(x)=L  ⟺  lim⁡x→a−f(x)=lim⁡x→a+f(x)=L\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L

Continuity

A function is continuous at x = a when the limit exists, the function is defined there, and they are equal.

lim⁡x→af(x)=f(a)\lim_{x \to a} f(x) = f(a)

Types of Discontinuity

Removable means the limit exists but the function is undefined or has a different value there. Jump means the left and right limits differ. Infinite means the function blows up.