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jostpuur
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Haha. This makes me happy. I wasn't the only one making the mistake 
A question professor couldnt solve
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SUMMARY
The discussion centers on the mathematical question of whether it is possible for the limit of the derivative of a function, as it approaches a point \( a \), to equal a value \( D \) while the derivative at that point \( f'(a) \) exists but is not equal to \( D \). Participants conclude that such a function cannot exist, referencing the Darboux property, which states that if a function is differentiable, its derivative must have the intermediate value property. The example \( f(x) = \frac{x+1}{x+1} \) is discussed, but it is clarified that the limit does exist as \( x \to -1 \), while \( f'(-1) \) is undefined, reinforcing the conclusion that the original question cannot be satisfied.
PREREQUISITES- Understanding of limits and derivatives in calculus
- Familiarity with the Intermediate Value Theorem
- Knowledge of the Darboux property of derivatives
- Basic proficiency in piecewise functions and their limits
- Study the Darboux property and its implications for differentiability
- Explore examples of piecewise functions and their limits
- Learn about the Intermediate Value Theorem and its applications in calculus
- Investigate counterexamples in calculus where derivatives do not exhibit expected properties
Mathematics students, calculus instructors, and anyone interested in advanced calculus concepts, particularly those exploring the properties of derivatives and limits.
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