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.
مثال کار شده · limit of sin(x)/x as x -> 0
قدم به قدم
- \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.
جواب رو نشون بده
حالا تو يه مساله انتخاب کن، يا خودت رو بنويس يا بکشي هر گام، يه تصوير، جواب مخفي ميشه تا وقتي که تو بپرسي
نمادهای استفاده‌شده در اینجا
هر نماد را برای تعریف کامل، تصویر و معنی هر حرف در آن بزنید.
چطور: 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.
سوالاتي که مردم ميپرسن
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.
بیشتر در Real Analysis
Sequences and their limitsConvergence of seriesImproper integralsTaylor approximation and errorFunctions of a complex variableSequences of functions and uniform convergence