Limit Calculator with Steps: L’Hôpital’s Rule & One-Sided

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Formula at a glance

Limit Calculator: L’Hôpital’s rule — If f/g → 0/0 or ∞/∞: lim f/g = lim f′/g′. Example: lim x→0 sin x / x → Limit: lim x→0 sin(x)/x = 1 — How the Limit Calculator works: L’Hôpital’s rule — If f/g → 0/0 or ∞/∞: lim f/g = lim f′/g′

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📘 About the Limit Calculator with Steps

A limit is the value a function approaches as x gets closer and closer to a point, even if the function is not defined there. Limits define continuity, derivatives and integrals, and describe behaviour at infinity (horizontal asymptotes).

The calculator tries direct substitution first. For 0/0 or ∞/∞ forms it applies L’Hôpital’s rule — differentiating the top and bottom — and shows each step. It also approaches the point numerically from both sides, so you can see when the left and right limits differ and the limit does not exist.

🧮 Formula

L’Hôpital’s rule

If f/g → 0/0 or ∞/∞: lim f/g = lim f′/g′

Famous limits

lim sin x/x = 1 (x → 0); lim (1 + 1/x)ˣ = e (x → ∞)

Existence

lim exists ⇔ left limit = right limit

✏️ Worked examples

lim x→0 sin x / x

You enter
Function f(x)	sin(x)/x
Variable	x
x approaches	0
Direction	Both sides (two-sided limit)

You get
Limit	lim x→0 sin(x)/x = 1
Left-hand limit (x→0⁻)	1
Right-hand limit (x→0⁺)	1

lim x→0 sin(x)/x = 1.

lim x→∞ (1 + 1/x)ˣ

You enter
Function f(x)	(1+1/x)^x
Variable	x
x approaches	inf
Direction	Both sides (two-sided limit)

You get
Limit	lim x→∞ (1/x + 1)^x = e ≈ 2.7182818

lim x→∞ (1/x + 1)^x = e ≈ 2.7182818.

🧭 How to use this calculator

What you enter

Function f(x)Use ^ for powers, * or nothing for ×: 3x^2 + 2sin(x), e^(2x), sqrt(x+1), ln(x).
VariableOne letter. Other letters are treated as constants.
x approaches
Direction

What you get

Limit
Left-hand limit (x→0⁻)
Right-hand limit (x→0⁺)

❓ Frequently asked questions

What does it mean when a limit does not exist?

The function does not settle on one value — the left and right limits differ (like |x|/x at 0), it grows without bound, or it oscillates (like sin(1/x)).

When can I use L’Hôpital’s rule?

Only for the indeterminate forms 0/0 and ∞/∞. Differentiate the numerator and denominator separately (not the quotient rule) and take the limit again.

How do I find a limit at infinity?

Divide by the highest power of x, or compare the growth of the top and bottom. Type inf for x → ∞.

What is lim (x→0) (1 − cos x)/x²?

1/2. Two applications of L’Hôpital’s rule give cos x/2 → 1/2.

→ Limit Calculator with Steps (L’Hôpital’s Rule)