What happens near a point?
Limits describe behavior as an input approaches a value, even when the function is not defined there.
MATHEMATICS · CHANGE & ACCUMULATION
Real Analysis & Calculus
Calculus studies change, approximation and accumulation. Real analysis gives those ideas a precise foundation through limits, continuity and rigorous reasoning.
THE CORE IDEAS
Three questions organize calculus.
How fast is something changing?
Derivatives measure instantaneous rate of change and the slope of a tangent line.
How much has built up?
Integrals measure accumulated quantity and signed area, and connect back to derivatives through the Fundamental Theorem of Calculus.
LEARNING PATH
Build calculus in the right order.
lim
Limits & continuity
One-sided limits, infinite limits, limit laws, continuity and the precise ε–δ definition.
Start with limits →f′Derivatives
Difference quotients, derivative rules, tangent lines, local approximation and applications.
Study derivatives →∫Integrals
Antiderivatives, definite integrals, Riemann sums and the Fundamental Theorem of Calculus.
Study integrals →ΣSequences & series
Convergence, infinite series, comparison tests and power series.
Explore convergence →∇fMultivariable calculus
Partial derivatives, gradients, multiple integrals and local approximation in several variables.
Go multivariable →∮Vector calculus
Line and surface integrals, conservative fields, Green's theorem, Stokes' theorem and divergence.
Connect local and global →BIG PICTURE
From local change to global geometry.
The path begins with functions and limits, builds derivatives and integrals, then extends them to several variables and vector fields. The major vector-calculus theorems unify these ideas as boundary–interior relationships.
Functions→Limits→Derivatives→Integrals→Multivariable→Vector calculus