The Precise Definition Of Limits In Mathematics: The Behavior Of Functions As Inputs Approach A Point

Precise definition of a limit (ALSO, precise definition is also known as delta – epsilon limit or ε – &).

if lim (x → a) f(x) = L thenfor every number ε > 0 and & > 0. then0 < |x - a| < & (x does not equal 3) and |f(x) - L| < ε Another way to write this is a - & < x < a + & and. L - ε < f(x) < L + ε

The precise definition of a limit is a mathematical concept that describes the behavior of a function as its input values approach a certain point, known as the limit point.

More formally, given a function f(x) defined on some interval that contains a limit point L, we say that the limit of f(x) as x approaches L is L (written as lim x→L f(x) = L), if for any positive number ε (epsilon), there exists a positive number δ (delta) such that if x is within a distance δ of L (i.e., |x-L| < δ), then the corresponding value of f(x) is within a distance ε of L (i.e., |f(x)-L| < ε). In other words, for any arbitrarily small positive number ε, we can find a corresponding positive value δ such that the distance between f(x) and L will be no more than ε when x is within a certain distance of L, including within δ distance. This definition ensures that the function value gets arbitrarily close to the limit value as x approaches L, without necessarily reaching it, and allows us to formalize concepts such as continuity, convergence, and differentiability.

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