Enter a function f(x), a point of approach, and choose your variable and direction — this Limit Definition Calculator evaluates the limit using the formal epsilon-delta definition. You get the limit value, the derivative via the difference quotient f'(x) = lim(h→0) [f(x+h)−f(x)]/h, and a step-by-step breakdown of the result.
Limit Value
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Derivative f'(a) via Difference Quotient
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Left-Hand Limit
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Right-Hand Limit
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Limit Exists?
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A limit describes the value a function approaches as its input approaches a given point. For example, lim(x→2) (x²) = 4 means that as x gets closer and closer to 2, f(x) gets closer to 4. Limits are the foundation of calculus, underpinning both derivatives and integrals.
The epsilon-delta definition states that lim(x→a) f(x) = L if, for every ε > 0, there exists a δ > 0 such that whenever 0 < |x − a| < δ, it follows that |f(x) − L| < ε. In plain terms, you can make f(x) as close to L as desired by keeping x sufficiently close to a.
The difference quotient is [f(x+h) − f(x)] / h, and taking its limit as h → 0 gives the derivative f'(x). This is the formal limit definition of the derivative and measures the instantaneous rate of change of f at a point.
A two-sided limit requires that f(x) approaches the same value L from both the left (x → a⁻) and the right (x → a⁺). A one-sided limit only considers approach from one direction. If the left-hand and right-hand limits are unequal, the two-sided limit does not exist.
A limit fails to exist when the left-hand and right-hand limits differ, when the function oscillates without settling on a value (like sin(1/x) near 0), or when the function grows without bound (diverges to ±∞). This calculator flags whether the two one-sided limits agree.
Common techniques include direct substitution, factoring and cancellation, rationalization, L'Hôpital's rule for indeterminate forms (0/0 or ∞/∞), and using limit laws (sum, product, quotient rules). This calculator uses numerical approximation via small delta-h steps to estimate the limit.
The primary output is the numerically approximated limit value L as x → a. The secondary outputs show the left-hand and right-hand limits separately, and the derivative f'(a) computed via the difference quotient. If left and right limits match (within tolerance), the limit exists.
The accuracy depends on the h step size you choose. A smaller h (e.g. 0.0001 or smaller) gives a more precise approximation of both the limit and the derivative. For well-behaved polynomial and rational functions, the results are accurate to several decimal places.