Limits and Derivatives

Limit of a Function

It describes the value that a function approaches as the input (x) gets closer and closer to a particular number, without necessarily equaling that number.

Algebra of Limits

Rules that allow us to break down complex limit problems into simpler operations, stating that the limit of a sum, difference, product, or quotient is the combination of their individual limits.

Standard Limits

Specific algebraic and trigonometric formulas, such as the limit of (x^n - a^n)/(x - a) as x approaches a, which help solve indeterminate forms directly.

Derivative from First Principles

The fundamental definition of a derivative using limits, representing the instantaneous rate of change of a function with respect to its variable, geometrically seen as the slope of the tangent.

Algebra of Derivatives

Standard rules including the sum rule, difference rule, product rule, and quotient rule used to differentiate combined functions efficiently without always resorting to first principles.