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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.

Àwọn ààtò àwọn ìṣàmúlò-ètò · 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}

Àwọn ìṣàmúlò-ètò

\lim_{x \to 0^+-} \frac{\sin{\left(x \right)}}{x}
Try direct substitution first.
\frac{0}{0}
Substitution gives 0/0: an indeterminate form.
\lim \frac{\cos{\left(x \right)}}{1}
L'Hôpital's rule: differentiate numerator and denominator separately.
= 1
Substitute.

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Bii o ṣe le: Continuity and differentiability, rigorously

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

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Real Analysis

Sequences and their limits Convergence of series Improper integrals Taylor approximation and error Functions of a complex variable Sequences of functions and uniform convergence