Limit Calculator

Calculate limits with clear steps: two-sided, one-sided, at infinity, and piecewise. Includes a table of values, DNE detection (jump / infinite / oscillation), and a mini graph preview.

Background

A limit asks what a function is “heading toward” as x approaches a value. If the left-hand and right-hand behavior agree, the two-sided limit exists — even if f(a) is undefined.

Enter inputs

Mode

Tip: Use Piecewise if your rule changes around the target point.

Function f(x) ?

Use x as the variable. Use ^ for powers (like x^2).

Approach value a

Infinity direction

This uses large x-values to estimate end behavior (dominant terms idea).

Piecewise definition (up to 3 pieces)

Define each piece by an interval and an expression. Example: piece 1: x<0 → 1/x, piece 2: x≥0 → x^2.

From

Right end

Expression

From

Left end

Expression

From

Ends

Expression

Use -inf and inf for ±∞. You can leave piece 3 blank.

Options

Round numeric values

Show step-by-step

Show table of values

Show mini graph preview

Quick picks

Chips fill values and solve immediately.

Result:

No results yet — enter inputs and click Calculate.

How to use this calculator

Pick a mode: point, one-sided, infinity, or piecewise.
Enter f(x) using x and functions like sin(), sqrt(), ln().
Enter a for point/one-sided/piecewise, or choose x → ∞ / x → −∞.
Use table + graph to sanity-check the numeric behavior near the approach.

How this calculator works

First it tries direct substitution (if finite, that’s the limit).
If that fails, it estimates left/right behavior using a sequence of shrinking steps.
It classifies results as finite, ±∞, or DNE (jump or oscillation).
A mini graph preview and table show the approach behavior visually.

Formula & Equation Used

A limit is written as:

$lim_{x \to a} f (x)$

If the left-hand and right-hand limits match, the two-sided limit exists:

$lim_{x \to a -} f (x) = lim_{x \to a +} f (x) = lim_{x \to a} f (x)$

Common algebra move (factor & cancel): If you get 0/0, try factoring and canceling a common factor.

$\frac{x^{2} - 1}{x - 1} = \frac{(x - 1) (x + 1)}{x - 1} = x + 1$

Conjugate/rationalizing move (classic square-root limit):

$\frac{\sqrt{x + 1} - 1}{x} \times \frac{\sqrt{x + 1} + 1}{\sqrt{x + 1} + 1} = \frac{(x + 1) - 1}{x (\sqrt{x + 1} + 1)}$

Standard trig limit (near 0):

$lim_{x \to 0} \frac{\sin (x)}{x} = 1$

Limits at infinity (rational functions): compare highest powers (dominant terms).

$If deg(numerator) < deg(denominator), then limit = 0$

$If degrees match, limit = \frac{leading coeffs}{leading coeffs}$

Example Problem & Step-by-Step Solution

Example 1 — 0/0 (factor & cancel)

Evaluate lim as x → 1 of (x^2 − 1)/(x − 1).

Direct substitution gives 0/0, so we simplify.
Factor: x^2 − 1 = (x − 1)(x + 1).
Cancel (x − 1) (for x ≠ 1): expression becomes x + 1.
Now substitute x = 1: limit = 2.

Example 2 — Rationalize a square-root fraction

Evaluate as x → 0: (sqrt(x+1) − 1)/x.

Substitution gives 0/0, so multiply by the conjugate.
Multiply top and bottom by sqrt(x+1) + 1.
Top becomes (x+1) − 1 = x, so the fraction simplifies to 1/(sqrt(x+1)+1).
Now substitute x = 0: limit = 1/(1+1) = 1/2.

Example 3 — Trig special pattern

Evaluate as x → 0: sin(x)/x.

This is a standard limit: lim_{x→0} sin(x)/x = 1.
So the limit is 1.

Example 4 — Two-sided DNE (jump)

Evaluate as x → 0 for the piecewise rule: f(x)=1 for x<0, and f(x)=2 for x≥0.

Left-hand limit: as x → 0−, f(x) → 1.
Right-hand limit: as x → 0+, f(x) → 2.
Since left ≠ right, the two-sided limit does not exist (DNE).

Frequently Asked Questions

Q: Can the limit exist if f(a) is undefined?

Yes. A limit is about what happens as x approaches a — the function value at a can be missing or different.

Q: When does a two-sided limit NOT exist?

Common reasons: left-hand and right-hand limits are different (jump), values blow up to ±∞, or the function oscillates without settling.

Q: Why use a table/graph?

They’re a quick sanity-check: you can see whether values are converging, diverging, or behaving differently from each side.

Q: What is the difference between a limit and f(a)?

The limit is what f(x) approaches as x gets close to a. f(a) is the actual function value at a (if it’s defined). They can be different.

Q: What should I do when substitution gives 0/0?

Try simplifying the expression: factor and cancel, rationalize (use a conjugate), or use a known trig limit (like sin(x)/x near 0) when applicable.

Q: How do limits at infinity work for rational functions?

Compare the highest powers of x. If the numerator degree is smaller, the limit is 0. If degrees match, it’s the ratio of leading coefficients. If the numerator degree is larger, the function grows without bound (±∞).