The discussion focuses on simplifying multivariable limits, specifically the limit of the expression $\displaystyle \lim_{x \to a} \dfrac{(x^n-a^n) - na^{n-1}(x-a)}{(x-a)^2}$. Participants utilize L'Hôpital's Rule to evaluate the limit, ultimately arriving at the result $\dfrac{n(n-1)(a^{n-2})}{2}$. Key points include the clarification that only the variable x approaches a, while a and n remain constants, and the correction of misconceptions regarding the factorization of $x^n - a^n$.
PREREQUISITESStudents of calculus, mathematics educators, and anyone seeking to deepen their understanding of limit evaluation techniques, particularly in the context of multivariable functions.
goody said:
No, there aren't. There are three symbols, x, a, and n. But only x is changing, it is going to a. a and n are constants. Only x is a variable.goody said:
No. x^n- a^n is NOT (x- a)(x^{n-1}- a^{n-1}. If you multiply the right side of that you get x^n- a^{n-1}x^n- ax^{n-1}+ a^n. x^n- a^n= (x- a)(x^{n-1}+ ax^{n-2}+ \cdot\cdot\cdot+ a^{n-2}x+ a^{n-1}.