Limits of Functions: The Hole and the Guarantee | Essence of Calculus

Can nearby outputs settle when a function has no value at the approach point? We begin with (x²−9)/(x−3), preserve its missing input, and turn a visual prediction into a complete epsilon–delta proof. A second proof for x² explains why a local bound and a minimum of two radii are useful. We distinguish values from limits, one-sided from two-sided approaches, and finite limits from unbounded behavior. The jump counterexample is justified by an explicit tolerance contradiction. The setting is real functions with accumulation points of their domains; the center is excluded from every limit test. Background: functions, algebra, absolute values, inequalities and sequence convergence. Essence of Calculus Chapters 0:00 A hole with a definite target 1:10 Factor the numerator 1:46 The hole remains 2:48 Request an output accuracy 3:56 A domain and an accumulation point 4:48 Prove the quotient limit 5:24 A nonlinear error factor 6:51 Two incompatible approaches 7:43 When the sides agree 8:04 Unbounded behavior near a point 8:24 Approach within the domain 8:41 Limits only depend on nearby values 9:00 Three values, one limit 9:37 From limits to continuity Continue exploring: Functions · the first lesson    • Functions: One Input, One Output | Essence...   References: OpenStax Calculus Volume1: The Precise Definition of a Limit: https://openstax.org/books/calculus-v...