The discussion centers on the commutation of limit and differential operators, specifically questioning whether the equation \(\lim_{x\to0} \left[\frac{d}{dy} \frac{d}{dx} f(x,y)\right] = \frac{d}{dy} \left[\lim_{x\to0} \frac{d}{dx} f(x,y)\right]\) holds for all functions \(f(x,y)\). It is established that this equality is valid if the mixed partial derivatives of \(f(x,y)\) are continuous at \(x=0\) for any given \(y\). The participants emphasize the practicality of differentiating first with respect to \(x\), applying the limit, and then differentiating with respect to \(y\) for complex functions.
PREREQUISITESMathematicians, calculus students, and anyone involved in advanced calculus or analysis who seeks to understand the interplay between limits and differentiation in multivariable functions.
No, not exactly. My f(x,y) is a huge compound function and it's a hassle to differentiate with respect to both variables and then take the limit, as opposed to differentiating by x, setting the limit and then differentiating (the now simpler function) by y.jedishrfu said:I think this is true provided the mixed partial derivatives of function f(x,y) are continuous at x=0 and whatever y=value.
http://www.math.ucsd.edu/~mradclif/teaching/Math10C/LectureNotes/second_order_partial_derivatives.pdf
Are you trying to prove this?