Limit Calculator | Evaluate Limits
Evaluate limits of functions numerically as x approaches a value from left, right, or both sides.
Supports: sin, cos, tan, exp, ln, sqrt, abs, ^
Common Limits Reference
| Expression | Value | Note |
|---|---|---|
| lim(x→0) sin(x)/x | 1 | Fundamental trig limit |
| lim(x→0) (1-cos x)/x | 0 | Cosine limit |
| lim(x→0) (eˣ-1)/x | 1 | Exponential limit |
| lim(x→0) (1+x)^(1/x) | e ≈ 2.718 | Euler's number definition |
| lim(x→∞) (1+1/x)ˣ | e ≈ 2.718 | Euler's number (alternate) |
| lim(x→0⁺) x·ln(x) | 0 | L'Hopital: 0·(-∞) form |
| lim(x→∞) xⁿ/eˣ | 0 | Exponential dominates power |
| lim(x→c) polynomial | f(c) | Direct substitution |
What Is the Limit Calculator | Evaluate Limits?
The Limit Calculator evaluates lim(x→a) f(x) numerically from both the left and right sides. It detects whether the two-sided limit exists, identifies infinite limits (vertical asymptotes), and evaluates limits at ±∞ (horizontal asymptotes). For indeterminate forms (0/0, ∞/∞, 0·∞), it applies L'Hôpital's Rule automatically.
- ›Two-sided limit: evaluates f(x) from both sides of a and checks if they agree to within numerical precision.
- ›One-sided limits: separately compute the left and right limits for functions with jump discontinuities.
- ›Limits at infinity: evaluates f(x) as x → +∞ and x → −∞ to find horizontal asymptotes.
- ›Indeterminate forms: detects 0/0, ∞/∞, 0×∞, ∞−∞, 1^∞, 0⁰, ∞⁰ and applies L'Hôpital's Rule numerically.
- ›Continuity check: compares the limit value to f(a), if they differ, the function is discontinuous at a.
- ›Common limits table: reference panel showing sin(x)/x→1, (1+1/n)^n→e, and other standard results.
Formula
| Limit Type | Notation | Meaning |
|---|---|---|
| Two-sided | lim(x→a) f(x) | Left and right limits must agree |
| Left-sided | lim(x→a⁻) f(x) | Approach from values x < a |
| Right-sided | lim(x→a⁺) f(x) | Approach from values x > a |
| At infinity | lim(x→∞) f(x) | Behavior as x grows without bound |
| Infinite limit | lim(x→a) f(x) = ∞ | Function grows without bound near a |
How to Use
- 1Enter the function f(x) in the input field (e.g., sin(x)/x, (x^2-1)/(x-1), (1+1/x)^x).
- 2Enter the point a that x approaches. Use "Inf" for +∞ or "-Inf" for −∞.
- 3Select the direction: two-sided (default), left-only (x→a⁻), or right-only (x→a⁺).
- 4Press Calculate (or Enter) to see the numerical limit value, left/right limit comparison, continuity status, and L'Hôpital applications if needed.
- 5Toggle "Show approach table" to see a table of f(x) values as x gets closer to a from both sides.
- 6Use the preset buttons for common textbook limits (sin(x)/x, definition of e, squeeze theorem examples).
- 7Click Reset (or Escape) to clear all fields.
Example Calculation
Example 1, Indeterminate form 0/0
Evaluate lim(x→1) (x² − 1)/(x − 1):
Example 2, Limit at infinity
Evaluate lim(x→∞) (3x² + 5x)/(2x² − 1):
Example 3, Famous limit defining e
Understanding Limit | Evaluate Limits
What Is a Limit?
A limit describes the value that a function approaches as the input gets arbitrarily close to a given point, even if the function is not defined at that exact point. The formal epsilon-delta definition captures this intuition precisely: for every desired accuracy ε, we can find a neighborhood δ around a such that f(x) stays within ε of L. This definition, introduced by Cauchy and Weierstrass in the 19th century, put calculus on rigorous mathematical foundations.
Limits are the foundation of all of calculus. The derivative is defined as a limit of a difference quotient: f'(x) = lim(h→0) [f(x+h) − f(x)]/h. The integral is defined as a limit of Riemann sums. Continuity is defined in terms of limits. Without limits, none of calculus exists.
Indeterminate Forms and L'Hôpital's Rule
When direct substitution gives an undefined expression, the limit may still exist, these are called indeterminate forms. The seven indeterminate forms are: 0/0, ∞/∞, 0·∞, ∞−∞, 0⁰, 1^∞, ∞⁰.
L'Hôpital's Rule handles 0/0 and ∞/∞ forms: if lim f/g gives 0/0 or ∞/∞, then lim f/g = lim f'/g' (provided the latter limit exists). The rule can be applied repeatedly if needed. Other indeterminate forms are converted to 0/0 or ∞/∞ first:
- ›0·∞ → rewrite as f = 1/(1/f) to get 0/0 form
- ›∞−∞ → combine fractions or rationalize to get 0/0 form
- ›0⁰, 1^∞, ∞⁰ → take the logarithm first: ln(f^g) = g·ln(f), then handle the resulting 0·∞ or 0/0 form
Types of Discontinuities
- ›Removable discontinuity: the two-sided limit exists but does not equal f(a), or f(a) is undefined. Example: f(x) = (x²−1)/(x−1) at x=1. The "hole" can be filled by redefining f(1) = 2.
- ›Jump discontinuity: left and right limits both exist but are not equal. Example: floor function at integers, or piecewise functions with mismatched pieces.
- ›Infinite discontinuity: one or both one-sided limits are ±∞. Example: f(x) = 1/x at x = 0. Indicates a vertical asymptote.
- ›Oscillatory discontinuity: the function oscillates infinitely without approaching any value. Example: sin(1/x) as x → 0.
Important Standard Limits
- ›lim(x→0) sin(x)/x = 1, basis for trig derivatives
- ›lim(x→0) (1−cos(x))/x = 0, second trig limit
- ›lim(x→∞) (1 + 1/x)^x = e, definition of Euler's number
- ›lim(x→0) (eˣ − 1)/x = 1, basis for exponential derivatives
- ›lim(x→0) ln(1+x)/x = 1, basis for logarithm derivatives
Frequently Asked Questions
What is the difference between a limit and a function value?
The limit lim(x→a) f(x) = L describes the value f(x) approaches as x gets arbitrarily close to a, but x never actually equals a. The function value f(a) is what the function equals at exactly x = a.
For continuous functions, these are the same: lim(x→a) f(x) = f(a). But for functions with holes, jumps, or asymptotes, they can differ, or f(a) may not even be defined. For example, f(x) = (x²−1)/(x−1) is undefined at x = 1, yet lim(x→1) f(x) = 2. This distinction is the core insight that makes limits more powerful than simple substitution.
When does a two-sided limit not exist?
A two-sided limit lim(x→a) f(x) fails to exist in three cases:
- ›Left limit ≠ right limit (jump discontinuity), e.g., the floor function at integers
- ›One or both one-sided limits are ±∞ (vertical asymptote), e.g., 1/x at x = 0
- ›The function oscillates without settling, e.g., sin(1/x) near x = 0
In all these cases, one-sided limits may still exist and be useful. The two-sided limit requires agreement from both sides.
How does this calculator detect indeterminate forms?
The calculator evaluates f(x) at the exact point a first. If this gives NaN, ∞, or an undefined arithmetic result (0/0, etc.), it flags the form as indeterminate and switches to numerical limit evaluation.
The numerical method approaches a from both sides with exponentially decreasing step sizes, using Richardson extrapolation to improve accuracy. For recognizable 0/0 patterns, it also attempts symbolic simplification, factoring differences of squares, rationalizing conjugates, to provide an exact answer where possible.
What is L'Hôpital's Rule and when can I use it?
L'Hôpital's Rule states: if lim f(x)/g(x) gives 0/0 or ∞/∞, and g'(x) ≠ 0 near a, then lim f(x)/g(x) = lim f'(x)/g'(x). The rule can be applied repeatedly if the result is still indeterminate.
Critical restriction: L'Hôpital's Rule can only be applied when the form is exactly 0/0 or ∞/∞. A common mistake is applying it to 0/2 (which simply equals 0) or ∞/2 (which simply diverges), the rule gives wrong answers in those cases. Always verify the indeterminate form before applying it.
What is the squeeze theorem?
The squeeze (or sandwich) theorem says: if g(x) ≤ f(x) ≤ h(x) near a, and lim g(x) = lim h(x) = L, then lim f(x) = L. The function f is "squeezed" to L by its bounds.
The most famous application: lim(x→0) sin(x)/x = 1. The proof shows cos(x) ≤ sin(x)/x ≤ 1 for small x using geometric area arguments, then since lim cos(x) = 1, the squeeze forces sin(x)/x → 1. This limit is the foundation for all trigonometric derivative formulas.
How do limits connect to derivatives?
The derivative f'(a) is defined as a limit: f'(a) = lim(h→0) [f(a+h) − f(a)]/h. This is a 0/0 indeterminate form that is resolved by algebraic manipulation, factoring, expanding, or simplifying the expression before taking h → 0.
Geometrically, this limit is the slope of the tangent line at x = a, obtained as the limit of secant slopes (connecting two nearby points) as the gap between them approaches zero. Every derivative rule in calculus (power rule, product rule, chain rule) is derived from this single limiting definition.
Can a limit equal infinity?
Yes, we write lim(x→a) f(x) = ∞ when f(x) grows without bound as x → a. Technically this means the limit does not exist as a finite number, but the notation tells you how it fails: it diverges upward. Similarly, lim = −∞ means it decreases without bound.
These infinite limits identify vertical asymptotes. Examples: 1/x at x = 0, 1/(x−2) near x = 2, tan(x) near x = π/2. For limits at x → ∞ (horizontal behavior), the result can be a finite number (horizontal asymptote) or ±∞ (unbounded growth).