Limit Calculator — L'Hôpital's Rule, Indeterminate Forms & Infinity
Evaluate limits of functions including limits at infinity, one-sided limits, and indeterminate forms. Covers L'Hôpital's rule, squeeze theorem, and an indeterminate forms table.
Limits are the foundation everything else in calculus is built on. Derivatives are defined as limits. Integrals are defined as limits. Without the limit concept, "instantaneous rate of change" is meaningless. Work through your limits with the CalcHub Limit Calculator.
What a Limit Says
lim(x→a) f(x) = L means: as x gets arbitrarily close to a (from either side), f(x) gets arbitrarily close to L. Note that f(a) doesn't need to equal L — or even need to exist.
Example: f(x) = (x² − 1)/(x − 1) At x = 1, this is 0/0 — undefined. But: (x² − 1)/(x − 1) = (x+1)(x−1)/(x−1) = x + 1 lim(x→1) (x²−1)/(x−1) = lim(x→1) (x+1) = 2Basic Limit Laws
If lim(x→a) f(x) = L and lim(x→a) g(x) = M:
| Law | Result |
|---|---|
| Sum/Difference | lim[f±g] = L ± M |
| Product | lim[f·g] = L·M |
| Quotient | lim[f/g] = L/M (if M ≠ 0) |
| Power | lim[f(x)ⁿ] = Lⁿ |
| Constant multiple | lim[c·f(x)] = c·L |
Indeterminate Forms
When direct substitution gives one of these forms, more work is needed:
| Form | Approach |
|---|---|
| 0/0 | Factor, rationalize, or L'Hôpital |
| ∞/∞ | L'Hôpital or divide by highest power |
| 0·∞ | Rewrite as 0/0 or ∞/∞ |
| ∞ − ∞ | Common denominator or rationalize |
| 1^∞ | Take logarithm |
| 0⁰ | Take logarithm |
| ∞⁰ | Take logarithm |
L'Hôpital's Rule
If lim(x→a) f(x)/g(x) gives 0/0 or ∞/∞, then:
$$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$
provided the right-hand limit exists.
Example: lim(x→0) sin(x)/x Direct substitution: 0/0 (indeterminate) Apply L'Hôpital: lim(x→0) cos(x)/1 = cos(0)/1 = 1 Example: lim(x→∞) x²/eˣ ∞/∞ form. Apply L'Hôpital twice: lim 2x/eˣ → still ∞/∞ → lim 2/eˣ = 0L'Hôpital can be applied repeatedly until the form resolves.
Limits at Infinity
Polynomial/Polynomial: Divide everything by the highest power of x. lim(x→∞) (3x² + 2x) / (5x² − 1) Divide by x²: (3 + 2/x) / (5 − 1/x²) As x → ∞: 2/x → 0 and 1/x² → 0 Result: 3/5 Guideline for rational functions:- If degree of numerator < denominator: limit = 0
- If degrees equal: limit = leading coefficient ratio
- If numerator degree > denominator: limit = ±∞
The Squeeze Theorem
If g(x) ≤ f(x) ≤ h(x) near a, and lim g(x) = lim h(x) = L, then lim f(x) = L.
Classic example: lim(x→0) x²·sin(1/x) We know −1 ≤ sin(1/x) ≤ 1, so: −x² ≤ x²·sin(1/x) ≤ x² Since lim(−x²) = lim(x²) = 0, by squeeze theorem: lim = 0One-Sided Limits
lim(x→a⁺) f(x) — approach from the right
lim(x→a⁻) f(x) — approach from the left
The two-sided limit exists only if both one-sided limits exist and are equal.
Example: f(x) = |x|/x From right: |x|/x = x/x = 1 From left: |x|/x = −x/x = −1 Since 1 ≠ −1, lim(x→0) |x|/x does not exist.Does L'Hôpital's rule always work?
L'Hôpital requires 0/0 or ∞/∞. It won't help with 0/3 (just substitute) or ∞ + 5 (already resolved). Also, repeatedly applying it can sometimes cycle — if you differentiate f/g and get g/f back, try a different method. Finally, L'Hôpital requires that the limit of f'/g' actually exists — if it doesn't, the rule is inconclusive.
What does it mean when a limit doesn't exist?
A limit fails to exist when: (1) the left and right one-sided limits differ, (2) the function oscillates without settling (like sin(1/x) near 0), or (3) the function grows without bound. Saying the limit "equals infinity" is technically a special case — it describes the behavior, but infinity is not an actual value in standard analysis.
How are limits used in defining continuity?
A function f is continuous at a if three conditions hold: f(a) exists, lim(x→a) f(x) exists, and they're equal: lim(x→a) f(x) = f(a). Continuity means you can draw the graph without lifting your pen — and it's the prerequisite for most of calculus to work.