Graphical Limit Estimator | Naruhodo Tutoring

Define the Limit

f(x) =

Use: x, Math.sin(x), Math.abs(x), Math.pow(x,n), Math.log(x), Math.sqrt(x)

x approaches a =

The value x is approaching (may be a point of discontinuity)

Quick Examples

Table of Values

Side	x	f(x)
Enter f(x) and a value to compute the table.

Limit Results

Results will appear here after you estimate the limit.

Graph

f(x)

x = a

Open circle (limit ≠ f(a))

Closed circle (f(a) defined)

How This Works — Step by Step

Choose your function and limit point. Enter any expression for f(x) and the value a that x is approaching. The function does not need to be defined at x = a — that's the whole point of limits!
Approach from the left. We plug in x-values that are close to a but slightly smaller: a−1, a−0.1, a−0.01, a−0.001, a−0.0001. As x gets closer to a from the left, f(x) stabilizes toward the left-hand limit L⁻.
Approach from the right. We plug in x-values slightly larger than a: a+0.0001, a+0.001, a+0.01, a+0.1, a+1. If f(x) stabilizes, that value is the right-hand limit L⁺.
Compare the one-sided limits. The two-sided limit lim(x→a) f(x) exists if and only if L⁻ = L⁺. If they differ (or one blows up to ±∞), the limit does not exist (DNE).
Check continuity. If f(a) is defined and equals the limit, the function is continuous at a. If f(a) is undefined but the limit exists, there is a removable discontinuity (a "hole"). If the limit itself doesn't exist, it's a non-removable discontinuity.
Use the graph to confirm. The canvas shows f(x) plotted near x = a. Use Zoom In to focus on the limit point. An open circle shows where the function is undefined; a closed circle shows the actual value f(a). Watch how the curve approaches (or doesn't approach) a single value.

Key Concepts

What Is a Limit?

A limit describes the value that f(x) approaches as x gets arbitrarily close to a — regardless of what happens at x = a itself. The function may be undefined there, yet the limit can still exist.

lim(x\toa) f(x) = L means: f(x) \to L as x \to a (but f(a) might \neq L, or not exist)

This is the foundational idea of calculus. Derivatives and integrals are both defined using limits.

One-Sided Limits & Discontinuities

When a function behaves differently on each side of a point, we use one-sided limits:

lim(x\toa⁻) f(x) — left-hand limit lim(x\toa⁺) f(x) — right-hand limit Two-sided limit exists ⟺ L⁻ = L⁺

Removable discontinuity: limit exists but f(a) ≠ L (a "hole").
Jump discontinuity: L⁻ ≠ L⁺ (step function).
Infinite discontinuity: f(x) → ±∞ (vertical asymptote).

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