Alphabeta MathDarboux, L'Hôpital, and Taylor's Theorem: Examples and Counterexamples
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✓ 8 results · all verified · 0 also independently AI-judged
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1 · Prerequisites
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- Topology of ℝ
2 · Logical flowchart
3 · Items
Example
A bounded C¹ periodic oscillator made from a quartic Hermite spline
Example
A differentiable function whose derivative is discontinuous
Example
For every k≥0, xᵏ|x| is Cᵏ but not Cᵏ⁺¹
Example
A function with positive derivative at 0 that is monotone on no neighbourhood of 0
Example
L'Hôpital evaluates lim_x→1(x³-x)/(x²-1) as 1
Counterexample
L'Hôpital's conclusion does not imply convergence of the derivative quotient
Example
The Taylor polynomial of (1-x)⁻¹ at 0 has the exact geometric remainder xⁿ⁺¹/(1-x)
Example