Calculus 1, part 1 of 2: Limits and continuity

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What you’ll learn

  • How to solve problems concerning limits and continuity of real-valued functions of 1 variable (illustrated with 491 solved problems) and why these methods work.
  • The structure and properties of the set of real numbers as an ordered field with the Axiom of Completeness, and consequences of this definition.
  • Arithmetic on the extended reals, and various types of indeterminate forms.
  • Supremum, infimum, and a reformulation of the Axiom of Completeness in these terms.
  • Number sequences and their convergence or divergence; the epsilon-definition of limits of sequences, with illustrations and examples; accumulation points.
  • Getting new limits from old limits: limit of the sum, difference, product, quotient, etc, of two sequences, with illustrations, formal proofs, and examples.
  • Squeeze Theorem for sequences
  • Squeeze Theorem for functions
  • The concept of a finite limit of a real-valued function of one real variable in a point: Cauchy’s definition, Heine’s definition; proof of their equivalence.
  • Limits at infinity and infinite limits of functions: Cauchy’s definition (epsilon-delta) and Heine’s definition (sequential) of such limits; their equivalence.
  • Limit of the sum, difference, product, quotient of two functions; limit of composition of two functions.
  • Properties of continuous functions: The Boundedness Theorem, The Max-Min Theorem, The Intermediate-Value Theorem.
  • Limits and continuity of elementary functions (polynomials, rational f., trigonometric and inverse trigonometric f., exponential, logarithmic and power f.).
  • Some standard limits in zero: sin(x)/x, tan(x)/x, (e^x-1)/x, ln(x+1)/x and a glimpse into their future applications in Differential Calculus.
  • Some standard limits in the infinity: a comparison of polynomial growth (more generally: growth described by power f.), exponential, and logarithmic growth.
  • Continuous extensions and removable discontinuities; examples of discontinuous functions in one, several, or even infinitely many points in the domain.
  • Starting thinking about plotting functions: domain, range, behaviour around accumulation points outside the domain, asymptotes (vertical, horizontal, slant).
  • An introduction to more advanced topics: Cauchy sequences and their convergence; a word about complete spaces; limits and continuity in metric spaces.
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