Chain Reaction 1!

Calculus Level 5

\[ f(x,y) = \begin{cases}\displaystyle \lim_{n\to \infty}\left(\lfloor x\rfloor^n+ y ^n\right)^{1/n} &
\text{if }x>y\\\displaystyle \lim_{n\to \infty}\left(x^n+ \lfloor y\rfloor^n \right)^{1/n} &
\text{if }y>x \end{cases}\]

\[g(a,b) = \begin{cases}\displaystyle \lim_{\substack{x\to a^+\\y\to b}} f(x,y) &
\text{if }x>y\\\displaystyle \lim_{\substack{x\to a\\y\to b^+}} f(x,y) & \text{if }y>x \end{cases}\]

\[h(x,m) = \begin{cases} 0 & \text{if } x<\lfloor m\rfloor\\\dfrac{(\lfloor m\rfloor-1)\lfloor m\rfloor(\lfloor m\rfloor+1)}3-\lfloor m-1\rfloor\lfloor m\rfloor^2& \text{if } x\ge\lfloor m\rfloor\end{cases}\]

\[\begin{align*} j(n, m) = &\dfrac1{h(1,m) + g(1,m)\cdot g(2,m)}\\ & \ \ \ \ \ +\dfrac1{h(2,m)+g(1,m)\cdot g(2,m)+ g(2,m)\cdot g(3,m)}\\ & \ \ \ \ \ \ \ \ \ \ +\dfrac1{h(3,m)+g(1,m)\cdot g(2,m)+ g(2,m)\cdot g(3,m)+ g(3,m)\cdot g(4,m)} \\\\ & \qquad{}\ \ \ \ \ \ \ +\cdots (n \text{ terms }) \end{align*}\]

Let the 4 functions \(f(x,y)\), \(g(a,b) \), \(h(x,m) \) and \(j(n,m) \) be defined as shown above.

Find \( \displaystyle \lim_{\substack{x\to\infty\\y\to e}} j(x,y) \).


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