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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.
דוגמה עובדת · limit of sin(x)/x as x -> 0
צעד אחר צעד
- \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.
גלה את התשובה
עכשיו אתה בחר בעיה, או הקלד או צייר בעצמך. כל צעד, תמונה, התשובה חבויה עד שתשאל.
סמלים המשמשים כאן
הקש כל סמל להגדרה המלאה, תמונה, ומה כל אות בזה אומר.
איך: 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.
שאלות שאנשים שואלים
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.
יותר בפנים. Real Analysis
Sequences and their limitsConvergence of seriesImproper integralsTaylor approximation and errorFunctions of a complex variableSequences of functions and uniform convergence