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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.

Mfano wenye matokeo · 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}

Hatua kwa hatua

  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.

Lafunua jibu
1

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Jinsi ya Kufaulu: 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.

Maswali ambayo watu huuliza

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.

Mengi zaidi katika Real Analysis

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