maths.free › Real Analysis › Continuity and differentiability, rigorously

Continuity and differentiability, rigorously

ε, δ continuity, the intermediate and extreme value theorems, the mean value theorem.

f is continuous at a when the limit equals the value; on a closed interval that buys the intermediate value theorem and a maximum. The mean value theorem says some tangent is parallel to the chord, the fact behind every "if f′ > 0 then f increases". Picture it: the chord and the parallel tangent. Think it: these theorems are where compactness and connectedness of [a, b] do their work.

Radni primjer · limit of sin(x)/x as x -> 0

Calculus · Limits

Limit of sin(x)/x as x → 0

\lim_{x \to 0} \frac{\sin{\left(x \right)}}{x}

Korak po korak

  1. \lim_{x \to 0^+-} \frac{\sin{\left(x \right)}}{x}

    Try direct substitution first.

  2. \frac{0}{0}

    Substitution gives 0/0: an indeterminate form.

  3. \lim \frac{\cos{\left(x \right)}}{1}

    L'Hôpital's rule: differentiate numerator and denominator separately.

  4. = 1

    Substitute.

Otkrij odgovor
1

Sada ti Izaberite problem, ili upišite ili nacrtajte svoj, svaki korak, slika, odgovor skriven dok ga ne pitate.

Neka tvoja radi.

Besplatni račun dodaje bilješke na svaku lekciju, zapis onoga što ste završili, vaše riješene probleme na jednom mjestu, i tutora kojeg možete pitati o ovoj stranici. Sama matematika je otvorena za svakoga, prijavljenog ili ne.

Prijavite se Prijava

Simboli korišteni ovdje

Dodirnite bilo koji simbol za potpunu definiciju, sliku i što svako slovo u njemu znači.

Kako: Continuity and differentiability, rigorously

  1. Try direct substitution first.
  2. Substitution gives 0/0: an indeterminate form.
  3. L'Hôpital's rule: differentiate numerator and denominator separately.
  4. Substitute.

Pitanja koja ljudi postavljaju

What is the ε, δ definition actually saying?

That you can make the output as close to the limit as anyone demands (within ε) by keeping the input close enough (within δ). It replaces "approaches" with a challenge-and-response that can be checked.

Why does the harmonic series diverge when its terms go to zero?

Because the terms shrink too slowly: group them as 1/3 + 1/4 > 1/2, 1/5 + … + 1/8 > 1/2, and so on: infinitely many halves.

Više u Real Analysis

Sequences and their limitsConvergence of seriesImproper integralsTaylor approximation and errorFunctions of a complex variableSequences of functions and uniform convergence