Calculus 1, part 1 of 2: Limits and continuity

Assess the role of precalculus before calculus, offering a flexible path to start calculus now and revisit precalculus as needed, with focus on building blocks and limits, continuity, and differentiability.

Explore the essence of precalculus 1, from natural numbers and functions to limits, continuity, and proof techniques, preparing you for calculus with intuition and essential tools.

Explores precalculus two with polynomials and rational functions, explains how polynomials approximate more complex functions with Taylor and Maclaurin polynomials, and covers zeros, monotonicity, factorization, and rational inequalities.

Explore precalculus three trigonometry: sine, cosine, tangent and reciprocals and inverses, learn via the unit circle, pi, and trigonometric identities to solve equations and explore applications.

Discover the essence of precalculus part 4, covering monotonicity in power, exponential, and logarithmic functions, Euler's number e, binomial theorem, Pascal's triangle, and related growth and decay applications.

Invite students to ask questions and use the q&a feature to seek clarifications, highlighting the instructor’s readiness to answer, correct mistakes, and provide explanations for precalculus content.

Continue with section 3 as a progression from section 2, generalizing formulas via induction. Learn the induction-based method and recap precalculus formulas from trig, exponentials, logarithms, polynomials, and rational functions.

Learn the principle of mathematical induction, including the base case and induction step, to prove statements about natural numbers and generalize formulas using the domino analogy.

Unpack how associativity, commutativity, and distributivity generalize to sums and products of many terms, guiding reasoning from pairs to n-term expressions with intuitive proofs.

Apply induction to derive a general method for exponents, proving a^(n+m)=a^n a^m and extending the power rule to real exponents using the product and quotient rules for exponential functions.

Work through the second example to illustrate general method: prove log base a of the product of n numbers equals the sum of their logs by induction, starting at n=2.

Explore De Morgan's laws and their generalization to any number of statements using truth tables and induction; connect logic to set theory and union and intersection.

Learn a general induction-based method for associative binary operations, showing how f preserves operations under star and diamond, with applications to exponentials, logarithms, and functional equations.

Apply the general method to concrete formulas by identifying the star and diamond operations and f across exponential, logarithmic, De Morgan, and negation–complement examples.

Explore linear transformations between vector spaces, defined by additivity and homogeneity, and extend linearity to sums of any number of vectors via induction.

Explore optional future topics linking limits, differentiation, and linear operations within vector spaces, illustrating addition, scaling, and linearity for functions, polynomials, and integrals.

Explore the sigma notation and the binomial theorem through an induction-based proof, contrast harder induction steps with easier base cases, and review precalculus groundwork like Pascal's triangle and combinatorial proofs.

Generalize squaring the sum beyond two terms using sigma notation and induction; apply associativity and the distributive law to derive the square of a sum with many terms.

Apply an induction method for sums and products, where the left-hand side is a function of a sum or product, by grouping terms and extending from two-term to three-term cases.

Generalize the product rule to any number of functions using an abstract, induction-based method, treating the derivative as a linear operation and emphasizing associativity and distributive laws.

Learn two essential trigonometric formulas—sine of sum and cosine of sum—through a quick complex-numbers trick, showing how to memorize or derive them and extend to related identities.

Derive the cosine and sine of the sum of three angles from the two-argument formulas, then generalize to more arguments using the same method.

Explore how the sine and cosine sum and difference formulas generate many trigonometric identities, derive the Pythagorean identity, and obtain double-angle and half-angle formulas for sine and cosine.

Derive and prove the sum of the first n natural numbers using the formula n(n+1)/2, with a geometric illustration and a clean induction step, including Gauss's story.

Compare derivation and proof by induction for formula two of the sum of squares of the first n positive integers, with base case, induction step, and application.

Show how partitioning the interval [0,1] into n pieces and summing x^2 values approximates the area under y = x^2, approaching 1/3 as n grows, foreshadowing the Riemann integral.

Examine derivation versus proof by induction for the sum of cubes, and prove that sum_{k=1}^n k^3 equals (n(n+1)/2)^2, the square of the sum of first naturals.

Generalize the triangle inequality to any number of real terms by induction, using geometric and case-based proofs, and show the sum of limits equals the limit of the sum.

Explore bernoulli's inequality, proves it by induction, and apply the binomial theorem to (1+d)^n with d>-1, illustrating the last induction case in this section.

Explore an elementary lemma for the Stolz–Cesàro theorem. If k_i/n_i are bounded between A and B with positive n_i, then (sum k_i)/(sum n_i) stays between A and B.

Learn how this section clarifies the real numbers as a complete ordered field, introduces limits, distance, and basic continuity, with optional deeper topics for real analysis.

Explore how the theory of real numbers justifies practical computations and continuity through completeness, the ordered field structure, and key theorems like boundedness and the intermediate value theorem.

Trace the development of real numbers from natural numbers to complex numbers, explaining why we add integers, rationals, and irrationals, and how decimal expansions distinguish rational from irrational numbers.

Explore the axiomatic foundations of real numbers, including ordered fields and completeness, and see how natural, integer, rational, and real numbers are built via primitive notions.

Learn how real numbers form an ordered field and derive addition and multiplication rules directly from axioms, including zero uniqueness, cancellation, and substitution for equality.

Optional: derive addition rules from the real-number axioms, proving uniqueness of zero and additive inverses, defining the difference, and verifying cancellation and the opposite of sums.

Derive multiplication rules from axioms by proving properties: x times zero equals zero, the zero product property, and that a nonzero number's inverse is nonzero, using distributivity and neutral elements.

Derive multiplication rules directly from the field axioms, using distributivity and additive inverses to prove key equalities such as x times zero equals zero for real numbers.

Derives inequalities from the ordered field axioms, showing how to add and multiply them with positive factors and proving inverses, signs, and positivity via contradiction.

Explore fields such as R, Q, C, and Z_p, and how their addition and multiplication follow field axioms, noting order and completeness.

Explore how the fields Q(sqrt(D)) with non-square D form ordered extensions between Q and R, closed under addition, multiplication, and inverses.

Explore how absolute value defines distances between real numbers, apply the triangle inequality, and use open and punctured neighborhoods to understand limits and continuity.

Explore supremum, infimum, maximum, and minimum within bounded real sets, and learn how least upper bounds and greatest lower bounds underpin order, completeness, and calculus.

Reformulate the axiom of completeness in terms of supremum and least upper bound, and prove its equivalence, showing every non-empty bounded above set has a supremum.

Explore the existence of roots beyond ordered field axioms, reveal why completeness is necessary, and connect precalculus root definitions to a rigorous existence proof.

Explore natural numbers through practical counting and Peano-axiom theory, emphasizing zero, the successor, induction, and embedding into the real numbers with the number axis.

Prove the minimum principle for natural numbers: every non-empty subset has a least element, using an inductive set and the not q implies not p logic.

Construct integers, rationals, and real numbers from axioms using dedekind cuts to partition rationals into left and right sets, and illustrate sqrt(2) as irrational.

Explore the floor function, the greatest integer less than or equal to x, with notation and geometric intuition. Learn about existence and uniqueness proofs based on the axiom of completeness.

Explore the three equivalent properties following from the axiom of completeness, their equivalence, and how the archimedean principle and density of rationals establish archimedean fields.

Explore how rational numbers are dense in the real numbers, guaranteeing a rational between any two reals, and how irrational numbers are also dense, enabling counterexamples and discontinuity examples.

Demonstrate that in any Archimedean field, each element can be approximated below by rational numbers. Conclude that there exists up to isomorphism exactly one ordered and complete number field.

Explore supremum and infimum with open and closed intervals, determine minimum and maximum, and learn proofs of least upper bounds through practical exercises.

Visualize and compute supremum, infimum, maximum, and minimum for several sets indexed by natural numbers, using Manim animations and step-by-step calculations.

Prepare for subtleties in the definitions of limits and continuity, including domain issues and polynomials and rational functions context, and explore epsilon-delta conditions and how limits relate to function values.

Discover how leading themes connect calculus concepts—accumulation points, limits, and continuity—through recurring examples across sections of this course.

Explore accumulation points, cluster points, and isolated points through distance, open balls, and neighborhoods, and define the derived set and discrete sets with practical examples.

Explore accumulation points, isolated points, and the derived set for the real-number subset DF, using visualization and domain concepts; relate these to limits and continuity.

Explore connections among supremum, infimum, and accumulation points using epsilon neighborhoods. Show that if the supremum or infimum does not belong to A, it must be an accumulation point.

Explore sequences and their limits, revisiting precalculus concepts of arithmetic, geometric, and harmonic progressions, with a focus on limits and Cauchy sequences.

Study sequences now to gain a clear, sequence-based understanding of limits and continuity. Explore monotone convergence, Riemann integrals, and indeterminate forms to build a foundation for calculus two.

Define sequences as infinite lists of numbers with notation a_n and the infinity symbol, and compare them to set notation.

Explore how sequences act as functions from natural numbers to real numbers and define them by explicit formulas, recursion, pictures, or verbal descriptions, with arithmetic, geometric, and Fibonacci sequences.

Explain how to read the first 11 terms from closed formulas, observe alternating and even-odd sub sequences, and compare CN and DN through simple formula manipulations.

This lecture demonstrates deriving closed formulas for sequences from given terms, with examples and verification, including a_n = (n+1)/n and a_n = 1 + (-1)^n/n.

Explore guessing the formula for a sequence from a recursive description and proving it by induction, illustrated with a0=1, an+1=2an, yielding an=2^n.

Examine a recurrence with two starting points by guessing the formula and proving it with induction. Use a1=3, a2=7 and a3=3a2-2a1 to illustrate the process with two base cases.

Explore simplifying the formula for S_n, derive values for n = 1 to 5, and connect the result to sequences, series, and limits and continuity.

Explore the optional derivation of the Fibonacci closed formula from its recurrence, prove by induction with two base cases, and note the role of sqrt(5) in the explicit expression.

Explore how sequences correspond to functions on positive arguments, comparing linear and arithmetic progressions with exponential and geometric progressions, including domain considerations and limits of the analogy.

Explore bounded sequences by applying |a_n| ≤ m for all n, and examine examples like the nth prime, pi digits, and the sequence (-1)^n n to reveal infimum and supremum.

Compare monotone sequences and monotone functions, using simple criteria to determine increasing or decreasing behavior on discrete domains, and note where derivatives and converse failures arise.

Compare two ways of depicting sequences—as functions in a coordinate system and on a single axis—and connect accumulation points to convergence, divergence, and set behavior.

Practice expressing that infinitely many terms of a sequence lie within an epsilon neighborhood of L. Demonstrate that for every epsilon > 0, almost all terms lie inside the neighborhood.

Explore accumulation points of sequences, and contrast them with accumulation points of sets, highlighting when they coincide or differ, and illustrating with examples like the alternating -1, 1 sequence.

Construct sequences with a specified number of accumulation points, from one to infinitely many, and demonstrate that rational numbers can make every real number an accumulation point.

Explore subsequences with informal and formal definitions, showing how index functions g and k_n form subsequences a_{k_n}; examine accumulation points, limits, and horizontal shifts.

Learn the definition and notation of the limit of a sequence, and distinguish convergent from divergent sequences with examples like constant sequences and a_n = n.

Ignore the first m elements and observe that as n grows, the sequence converges when tail terms enter any epsilon neighborhood of the limit.

Accumulation points are limits of subsequences of a sequence. Use epsilon neighborhoods to build subsequences that converge to the accumulation point; even indices tend to one, odd to minus one.

Learn that a convergent sequence has exactly one accumulation point, equal to its limit, and see why a single accumulation point does not guarantee convergence through epsilon reasoning and counterexamples.

Explore the limit of a sequence from the formal definition, using epsilons and floors to find n_epsilon; prove that 1/n → 0 and discuss epsilon neighborhoods.

Prove a limit from the epsilon definition by choosing n_epsilon so that a_n lies within epsilon of two thirds, illustrating convergence of the sequence.

Apply the epsilon definition to a rational sequence, estimate the expression, and prove that the limit equals two thirds as n grows large.

Compute the limit from the definition for the sequence a_n = 2 + 3^{-n}, showing it converges to 2 as n grows using epsilon arguments and logarithms.

Master epsilon proofs to unlock limits and continuity. Use the triangle inequality and absolute value to bound distances, relate neighborhoods, and learn epsilon-based tricks (R2, R3, R4).

Explore properties of convergent sequences: boundedness, uniqueness of the limit, and subsequences sharing that limit; and examine equivalences with distance to the limit, scaling, and products with bounded sequences.

Explore the squeeze theorem for sequences, which bounds A between two convergent sequences B and C that both tend to the same L.

Apply the squeeze theorem to sequences to prove q^n → 0 as n → ∞ for -1 < q < 1, using Bernoulli's inequality or the binomial theorem.

Explore the squeeze theorem for sequences by solving exercise 10: prove that the nth root of n and that of a positive constant converge to 1, using the binomial theorem.

Discover how limits behave under arithmetic operations on convergent sequences. Learn to apply the limit rules to sums, products, powers, quotients, and roots to motivate continuity.

Apply the limit laws to compute limits of sequences, using products, quotients, sums, and boundedness to verify convergence; practice includes exercises drawn from previous examples to illustrate the method.

Apply limit laws to derive new limits from old ones, evaluating sums, products, and quotients by factoring out powers, with bounded sequences like sin n and 1/√n tending to zero.

Derive new limits from old limits using geometric progressions and dividing by the dominant power, then apply conjugate tricks to simplify limits of sequences and products tending to zero.

Proving new limits from old limits, this lecture uses the squeeze theorem and induction to show sums, products, and powers of convergent sequences equal their limits.

Show that the limit of the reciprocal and the quotient of two convergent sequences equal the reciprocal and quotient of their limits, with nonzero denominators, via the squeeze theorem.

Show how the k-th root of a_n tends to the k-th root of a as a_n converges to a, covering even and odd k cases with a three-case squeeze argument.

Prove that if a_n tends to a, then |a_n| tends to |a|, and show a counterexample where |a_n| converges but a_n diverges using the alternating sequence (-1)^n.

Explore the Weierstrass theorem on convergence of bounded monotone sequences in real numbers. See how increasing bounded sequences converge to their supremum and decreasing bounded sequences converge to their infimum.

Apply Weierstrass' theorem to the sequence a_n=(1+1/n)^n, prove its monotonic increase via Bernoulli's inequality, and use the limit to confirm the definition of e.

Explore Weierstrass theorem with two sequences: a geometric sequence with 0<q<1 converging to 0, and a nested square-root sequence converging to 2, via monotone and bounded steps.

Explore monotone, bounded, and convergent sequences through a guided test, with examples and counterexamples illustrating accumulation points, sums, quotients, and key properties P1 and T1.

Introduce extended reals by adding plus infinity and minus infinity, and define their neighborhoods. Explore how sequences diverge to infinity and their accumulation points in the extended framework.

Explore how arithmetic on extended reals remains continuous by defining binary operations with convergent sequences and proving continuity for addition, multiplication, division (zero restrictions) and powers.

Extend arithmetic to extended reals by defining addition, subtraction, multiplication, and division with infinity, yielding a continuous extension with intuitive rules and accompanying proofs.

Extend arithmetic to extended reals and establish continuity of taking powers. Learn power rules for positive bases, reciprocals, roots, and fractional exponents.

Explore how extended real arithmetic rules determine limits of sequences defined by polynomials and rational functions, including leading coefficient behavior and plus or minus infinity.

Extend real numbers to extended reals and define arithmetic, then explain indeterminate forms, showing how infinity minus infinity, zero times infinity, and related cases resist unique limits.

Explore how infinities compare in calculus, focusing on indeterminate forms like infinity over infinity, and learn to use the fastest growing term to evaluate limits in exercise 16.

Explore the limit used to compare infinities in the exponential function, and prove a^n/n! tends to zero; then build e^x via Maclaurin series, with two proofs: Weierstrass and squeeze.

Explore how infinities compare with more quotients, proving when reciprocals diverge or vanish using Weierstrass theorem, sequences, and exponential growth concepts.

Explore optional Cauchy sequences in metric spaces and completeness; learn how real numbers arise from rationals via Cauchy sequences and the role of convergence and distance axioms.

Explore proximity in limits of functions at a point through introductory examples and the epsilon-delta intuition, linking to continuity in calculus.

Explain how limits are meaningful at accumulation points of a function's domain, including boundary points, where x approaches a and values get arbitrarily close to L.

Compare two functions f and g: domains, ranges, graphs. G is y = x + 1; f omits x = 1 and has limit 2 as x approaches 1.

Explore limits using f(x)=(x-1)/(x^2-1) and its cancellation to 1/(x+1), exclude x=±1, reveal a hole at (1,1/2), and show the limit as x approaches 1 is 1/2.

Examine the standard limit sin x over x as x approaches zero, its domain exclusion, and its role in motivating the derivatives of sine and cosine in calculus.

Examine the precalculus example with e^x showing that (e^x - 1)/x tends to 1 as x approaches 0, an indeterminate 0/0 form, foreshadowing the derivative of e^x.

Formally define the limit of a function at a point with the epsilon-delta condition, where f(x) tends to L as x tends to a, an accumulation point of the domain.

Demonstrates the epsilon-delta proof that a limit of a real-valued function near a point is unique, using two candidate limits and the triangle inequality.

Apply the definition to constant functions, showing that the limit as x tends to a equals c and that such functions are continuous, illustrated with a graph.

Apply the epsilon-delta definition to f(x)=x and show that as x approaches any real a, the limit equals a, proving continuity.

Demonstrates the limit of f(x) as x approaches nine by applying the epsilon-delta definition, simplifying f to the square root of x plus three, and concluding delta equals three epsilon.

Explore examples where the limit does not exist, including the floor function at integers, the rational numbers' characteristic function on [0,1], and piecewise graphs.

Introduce left and right limits and define one-sided limits using left and right neighborhoods. Explain when a limit exists if the two one-sided limits agree, with graph-based examples.

Delve into one-sided limits with floor and piecewise examples, visualize left and right limits with the graphic method, and note epsilon-delta proofs for key limits.

Explore one-sided limits using the signum function and its square, compare left and right limits at zero, and identify discontinuity when the two-sided limit fails.

Explore the theorem of new limits from old limits for real-valued functions, focusing on linear combinations, products, and quotients near a. See accompanying lemmas and epsilon-delta proofs.

Apply the limits from old limits theorem to form new limits using sum, difference, product, and quotient. Study problems that show powers and graph readings to reinforce these limit rules.

Learn a visual, intuitive approach to limits, including epsilon-delta concepts, using sequences and graphs, illustrated by 12 Manim animations connecting x tending to a point with function values.

Explore Heine's sequential definition of limits and notation, and use sequences like floor x and sine of one over x to show nonexistence of limits at a and zero.

Showcases the equivalence of Cauchy's epsilon-delta and Heine's sequential definitions of the limit at an accumulation point, with a detailed indirect proof where the limit is L.

Discover two proofs of the limits theorem by reducing function cases to sequence limits, using Heine's definition and epsilon-delta reasoning, and learn the advantage of studying sequences first.

Explore the meaning of continuity by linking the epsilon-delta definition to limits and function values at accumulation points, including isolated points, and continuity of sums, products, and quotients on intersections.

Use a flowchart to determine continuity at a by checking domain, accumulation point, and the limit, then compare the limit to f(a).

Explore continuity and discontinuity in real-valued functions by examining left and right limits, domain considerations, and examples like constant, identity, floor, and piecewise functions.

Compare the calculus definition of continuity—f(a) defined, limit exists, and limit equals f(a)—with other definitions, noting topology relevance and removable or jump discontinuities.

Explore the continuity of elementary functions—polynomials, rational functions, and power functions—and learn how limits, domain, and limit laws prove continuity in these basics.

Explore the continuity of trigonometric functions—sine, cosine, tangent, cotangent, secant, and cosecant—using epsilon-delta proofs, unit circle intuition, and domain considerations.

Explore a key lemma on one-sided limits of monotone functions, proving the existence of left and right limits at interior points and illustrating jump discontinuities and continuity via supremum arguments.

Proves the continuity of f(x)=a^x for any positive base a ≠ 1 by showing left and right limits coincide via monotonicity and rational approximations; connects to hyperbolic sine and cosine.

Explore how the composition of continuous functions is continuous, and follow a two-function epsilon-delta proof using domain inclusion and continuity at a and f(a).

Apply the theorem that the composition of continuous functions is continuous through three examples. Analyze continuity at zero for extended and product functions, including x sin(1/x), using an epsilon-delta bound.

Show that a continuous invertible monotone function has a continuous inverse that is strictly increasing or decreasing. It maps intervals to intervals and lacks jump discontinuities.

Illustrates why the interval assumption matters for the continuity of inverse functions by presenting a piecewise function on disjoint intervals, revealing a jump discontinuity in the inverse.

this lecture derives continuity consequences for inverse functions, confirming arcsin, arccos, arctan, and logarithm are continuous, and shows x^x via e^{ln x^x} is continuous and a_n^{b_n} -> a^b.

Explore strategies for converting indeterminate zero by zero limits into determinate forms using factoring, conjugates, and algebraic cancellation. Practice problems illustrate standard limits and problem solving.

Use the squeeze theorem for functions to show f(x) tends to L when g(x) and h(x) tend to L near a, via delta neighborhoods and corollaries, e.g., x sin(1/x).

Apply the squeeze theorem to f(x) = sin x/x, bounding it between cos x and 1 using triangle and sector. Conclude sin x/x tends to 1 as x approaches 0.

Divide the disk into n equal triangles and compute each area, then take the limit as n grows, using sin x/x tends to 1, to derive a = pi r^2.

Explore two standard limits as x tends to zero: (e^x-1)/x→1 and log(1+s)/s→1, using exponential and logarithm inverses, the power rule, and a change of variables.

Practice solving limits by applying standard limits in zero, variable substitution, and algebraic cancellations to handle sine, logarithm, and tangent expressions.

Learn to solve zero limits with trigonometric functions by rewriting tan x as sin x over cos x, using conjugates, and applying sin x / x and continuity.

Solve two trigonometric limits by changing variables, using cofactor identities and factoring to resolve zero by zero indeterminate forms, then apply standard sine, cosine, and tangent limits.

Learn to solve standard limits in zero by transforming indeterminate forms with e, sine, and arcsin, using lemmas to reveal limits such as e and arcsin t over t.

Solve standard limit problems by multiplying by the conjugate to resolve indeterminate forms. Apply the difference of squares and cancel factors, then plug in the limit to obtain determinate values.

Highlight continuity's role in solving limits via sums, products, quotients, and compositions. Note that piecewise functions may fail continuity, influencing limit behavior as x approaches 1 or 0.

Explore one-sided limits with absolute value, analyze left and right limits of sine x over x near zero, and use a variable change to compute subtle limits.

Analyze right and left limits of a composed function with polynomials x^3−x and x^2−x^4 using sign analysis to determine when the inner argument is negative or positive.

Explore infinite limits with the graph of y = 1/x, showing horizontal and vertical asymptotes. Examine right and left limits at zero tending to plus and minus infinity.

Discover that extended reals underpin limits for sequences and functions, including infinity and indeterminate forms. Explore examples and animations that introduce infinite limits with concise definitions.

This lecture shows that monotone functions on positive reals have limits at infinity equal to the sequence limits, enabling reuse of sequence results with examples from exponential and power functions.

Learn the formal definition of infinite limits as x tends to infinity, compare to improper limits of sequences, and explore the four possible epsilon-delta style scenarios with extended reals.

Explore infinite limits at infinity using the sequential condition and the epsilon-delta definition, and apply to polynomials, power functions, and exponentials on the positive axis.

Explore finite limits at infinity using epsilon-delta definitions for plus and minus infinity, and analyze f(x)=x sin(1/x) to show the limits are 1 and explain the continuous extension at zero.

Use the sequential condition to define finite limits as x tends to infinity, with xn → ∞ and f(xn) → L, including plus and minus infinity and extended real arithmetic.

Examine horizontal asymptotes and limits at infinity for functions and rational expressions, including degree comparisons and lines y = a for plus and minus infinity.

Examine infinite limits at accumulation points of the domain outside the domain, using delta neighborhoods and a large E to show f(x) grows without bound as x approaches a.

Examine infinite limits at accumulation points using a sequential condition, focusing on rational functions and polynomials near zero, where the lowest exponents dominate and left-right limits matter.

Examine vertical asymptotes through infinite left and right limits, accumulation points of the domain, and zeros of denominators in rational functions, including cancellations that remove asymptotes and create holes.

Tangent and arctangent graphs reveal vertical and horizontal asymptotes: tangent has vertical asymptotes at x pi/2 plus k pi, while arctangent has horizontal asymptotes at y plus minus pi/2.

Explore limits as x tends to infinity, compare growth rates of exponential and logarithmic functions, and prove that a^x/x tends to infinity for a>1 by extending sequence results to functions.

Shows a function with two different horizontal asymptotes: y = 1/2 as x tends to plus infinity and y = −1/2 as x tends to minus infinity.

Master limits by applying conjugates and the difference of squares, then resolve infinity and indeterminate expressions by canceling the quickest growing term to a finite value.

Explore solving a second limit problem by evaluating inner limits inside arctan and arccos, using continuity and careful handling of x/(1+x^2) as x→−∞ and related expressions as x→∞.

Apply limit techniques to infinity, transforming expressions toward the standard (1+a_n)^{1/a_n} form that tends to e, using logarithms and power rules.

This lecture shows that the limit of x^x as x approaches zero from the right equals one, using a lemma on x^α log x and setting t to 1/x.

Show how limits do not exist by constructing divergent sequences with tangent and sine as inputs tend to infinity, then use the tangent addition formula to reach a contradiction.

Analyze two limits as x approaches 2: one equals minus infinity, the other does not exist due to divergent one-sided limits, illustrating cancellation and a vertical asymptote in rational functions.

Explore the three types of discontinuities—removable, jump, and infinite. Define a continuous extension by setting f(a) to the common left-right limit when they agree.

Determine the value of k that makes the piecewise function continuous at x = 2 by equating the left limit to the function value, yielding k = 4.

Problem 2 shows a removable discontinuity at x = 0; the limit is 3 while f(0) = 2, so set f(0) = 3 for continuity on x > -1/3.

Demonstrate extending the function by canceling x-1 after long division, compute the limit at x→1, and set the extension value to 0 since the limit exists.

Determine a and b so f(x) has limits 3 at zero and π at infinity, yielding b = 1 and a = 2 - 2/pi, using sin(2x)/x and arctan x.

Compute a piecewise area-based function from three rectangles and demonstrate continuity by matching left and right limits at each junction, resulting in a globally continuous function on nonnegative reals.

Prove that the limit of the nth root of a^n plus b^n equals the larger of a and b, apply to f(x)=lim nth root of (1+x^2)^n, yielding f(x)=max(1,x^2) and continuity.

Solve problem eight on continuity and discontinuities. Derive a piecewise f(x) with zeros where sin x > 1/2 or < -1/2, and jumps at sin x = ±1/2.

Explore functions with different numbers of discontinuity points, including floor function and Toma function. Learn how piecewise linear constructions create jumps and how accumulation points affect continuity.

Learn how to build piecewise functions from a sum of absolute values and ensure the ends meet, using calculus continuity to guarantee a continuous function on real numbers.

Explore the properties of continuous functions on closed intervals, including boundedness, the max-min theorem, and the intermediate value theorem, with practical examples.

Demonstrates the separation lemma for a function continuous at an accumulation point, producing deltas that bound f(x) above and below by mu and ni near a.

Prove the boundedness theorem for continuous functions on a closed interval by contradiction, using sets s_k and their infima to establish bounded above and below.

the max-min theorem shows that a continuous function on a closed interval attains both its maximum and minimum, tied to supremum and infimum of its range.

Identify the midpoint of the interval [a, b] as a plus half the distance, namely a + (b - a)/2 = (a + b)/2.

This lecture explains the intermediate value theorem for a continuous function on [a,b], showing there exists c with f(c)=d whenever d lies between f(a) and f(b).

Explore how continuous function properties and the intermediate value theorem determine ranges of polynomials, trigonometric, exponential, and logarithmic functions, including compositions and domains.

Demonstrate that any continuous function f mapping [0,1] into [0,1] has a fixed point x0 with f(x0)=x0, using the intermediate value theorem on g(x)=f(x)−x.

Use the intermediate value theorem on continuous f and g on [a,b], with f(a)<g(a) and f(b)>g(b); the function h(x)=f(x)−g(x) has a zero, so f(x)=g(x) at some x in (a,b).

Apply the intermediate value theorem to cubic p(x) = x^3 - 3x + 1, using sign changes at -2, -1, 0, 1, 2 to locate zeros in (-2,-1), (0,1), (1,2).

Explore properties of continuous functions on the interval [a, ∞), proving boundedness when the limit at infinity is finite using epsilon-delta and the boundedness theorem.

Show none of these four sets can be the range of a continuous function on a closed interval, by max-min, boundedness, and the intermediate value theorem.

Explore properties of continuous functions, including boundedness on intervals and limits, behavior under sums and bijections, with counterexamples to sharpen intuition.

Explore uniform continuity, with a universal delta, compare to ordinary continuity, and illustrate via f(x)=2x+1 and g(x)=sqrt(x) as uniformly continuous, while f(x)=1/x is not.

Explore open, closed, compact, and connected sets in metric spaces, including interiors, boundaries, closures, and open balls, with real-number interval examples and a teaser for future generalizations.

Reformulate the boundedness theorem, the max min mean theorem, and the intermediate value theorem for metric spaces, linking continuous images to compactness and connectedness.

Explore three characterizations of continuity for functions between metric spaces: epsilon-delta, sequential, and the topological preimage criterion using open sets.

Learn to begin graphing real-valued functions of one real variable by determining domains, accumulation points, one-sided limits, discontinuities, and asymptotes, with a teaser for derivatives in the next course.

Explore slant asymptotes for rational functions and learn to compute their slope and intercept from limits as x approaches infinity or minus infinity.

Analyze the rational function f(x) = (x^2+2x+5)/(x+1). Determine domain (real numbers except -1), y-intercept 5, and no zeros; vertical asymptote at -1 and slant asymptote y = x+1.

Determine the domain and zeros of a function with square roots, and show horizontal asymptotes: y = 1/2 as x → ∞ and y = −1/2 as x → −∞.

Analyze f(x)=arctan(x/(x+1)) by noting its domain is real numbers excluding -1, y-intercept 0, and zero at x=0. Horizontal asymptotes occur at y=pi/4 for both directions; no vertical asymptote at -1.

Analyze the domain and asymptotes of a function with arcsin and tangent, showing a vertical asymptote at -1 and that near pi/2 - 1 the function tends to zero.

Study f(x)=exp(1/(1+x)) with a one-sided vertical asymptote at x=-1; left limit is 0, right limit is infinity, horizontal asymptote y=1, and the function is positive and decreasing on its domain.

Wrap up calculus 1, part 1 of 2, with an invitation to ask questions and a preview of part two on derivatives with applications.