Understanding Limits in Calculus: Defining Limits with Epsilon-Delta Notation

Limit definition

As x approaches c, (number/constant), as the values of x get arbitrarily close to c, the values of f(x) approach L, the limit (Y value on graph)

In calculus, a limit is a value that a function, sequence or series approaches as the input or index approaches a certain value. The limit definition is a formal mathematical statement that defines the concept of a limit.

The limit definition states that the limit of a function f(x) as x approaches a certain value, say c, is L if and only if for every positive number ε (epsilon), there exists a positive number δ (delta) such that if 0 < |x-c| < δ, then |f(x) - L| < ε. This can be written symbolically using the epsilon-delta notation as: lim [x → c] f(x) = L if and only if for every ε > 0, there exists a δ > 0 such that if 0 < |x-c| < δ, then |f(x) - L| < ε. Simply stated, this definition means that as we get arbitrarily close to the value c, the values of the function f(x) get arbitrarily close to the limit L. The value of f(x) need not be equal to L at c, but as x approaches c, the values of f(x) should be as close to L as we want them to be, provided we choose a small enough value of δ.

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