lim Limit Calculator
Evaluate limits numerically, explore one-sided limits, limits at infinity, and indeterminate forms with step-by-step solutions.
Evaluate lim f(x) as x → a
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Limit value
One-Sided Limits
Limit as x → ±∞
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Limit as x → ∞
Important Limit Rules & Theorems
Constant: lim[c] = c
Sum Rule: lim[f+g] = lim[f] + lim[g]
Product Rule: lim[f·g] = lim[f] × lim[g]
Squeeze Theorem: If g(x) ≤ f(x) ≤ h(x) and lim g = lim h = L, then lim f = L
Special: lim(x→0) sin(x)/x = 1
Special: lim(x→0) (eˣ−1)/x = 1
L'Hôpital: If 0/0 or ∞/∞, use lim f/g = lim f'/g'
Indeterminate Forms
When direct substitution gives 0/0, ∞/∞, 0·∞, or ∞−∞, use algebraic manipulation, factoring, or L'Hôpital's Rule to resolve.
How the Limit Calculator Works
How Limits Work
A limit describes what value a function approaches as the input approaches a specific point. Limits are fundamental to calculus and help us understand function behavior at critical points.
Mathematical Definition:
Formal Definition:
lim(x→a) f(x) = L means: for every ε > 0, there exists δ > 0 such that
if 0 < |x – a| < δ, then |f(x) – L| < ε
The precise mathematical definition
lim(x→a) f(x) = L means: for every ε > 0, there exists δ > 0 such that
if 0 < |x – a| < δ, then |f(x) – L| < ε
The precise mathematical definition
Intuitive Definition:
As x gets arbitrarily close to ‘a’, f(x) gets arbitrarily close to ‘L’
Easier to understand conceptually
As x gets arbitrarily close to ‘a’, f(x) gets arbitrarily close to ‘L’
Easier to understand conceptually
Types of Limits:
- Two-sided: lim(x→a) f(x) – exists only if left and right limits are equal
- Left-hand: lim(x→a⁻) f(x) – approaching from values less than a
- Right-hand: lim(x→a⁺) f(x) – approaching from values greater than a
- Infinite limits: Function approaches ±∞ (vertical asymptotes)
- Limits at infinity: Behavior as x approaches ±∞ (horizontal asymptotes)
Real-World Applications:
- Physics: Instantaneous velocity, acceleration at a specific moment
- Engineering: Stress analysis at critical points, failure thresholds
- Economics: Marginal analysis, optimization problems
- Medicine: Drug dosage limits, therapeutic windows
- Computer Science: Algorithm complexity analysis, convergence testing
Common Limit Types:
- Continuous functions: lim(x→a) f(x) = f(a)
- Removable discontinuity: Limit exists but ≠ f(a) (hole in graph)
- Jump discontinuity: Left and right limits exist but differ
- Essential discontinuity: At least one one-sided limit doesn’t exist
- Indeterminate forms: 0/0, ∞/∞ – require special techniques (L’Hôpital’s rule)