What you’ll learn

How to solve problems concerning limits and continuity of realvalued 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 epsilondefinition 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 realvalued 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 (epsilondelta) 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 MaxMin Theorem, The IntermediateValue 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^x1)/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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