A calculus problem by Rindell Mabunga

$\huge \lim _{ z\rightarrow \infty }\sum _{ x=0 }^{ \infty }[{ \sum _{ y=1 }^{ z }\frac { y^{x + 1} }{ x!(z^{x+2}) } }]$

$\huge =\lim _{ z\rightarrow \infty } \sum _{ y=1 }^{z}[\sum _{ x=0 }^{ \infty }{\frac { y^{x + 1} }{ x!(z^{x+2}) } }]$

$\huge =\lim _{ z\rightarrow \infty } \sum _{ y=1 }^{ z }[\sum _{ x=0 }^{ \infty }{(\frac{y}{z^2} \times \frac{(\frac{y}{z})^x}{x!}) }]$

$\huge =\lim _{ z\rightarrow \infty } \sum _{ y=1 }^{ z }[(\frac{y}{z^2})\sum _{ x=0 }^{ \infty }{(\frac{(\frac{y}{z})^x}{x!}) }]$

$\huge =\lim _{ z\rightarrow \infty } \sum _{ y=1 }^{ z}[(\frac{y}{z^2})(e^{\frac{y}{z}})]$

$\huge =\lim _{ z\rightarrow \infty } \sum _{ y=1 }^{ z }[(\frac{1}{z})(\frac{y}{z})(e^{\frac{y}{z}})]$

$\huge =\lim _{ z\rightarrow \infty }(\frac{1}{z}) \sum _{ y=1 }^{ z }[(\frac{y}{z})(e^{\frac{y}{z}})]$

$\huge = \int _{ 0 }^{ 1 }{ y{ e}^{ y }dy }$

$\huge = ye^y - e^y]_{0}^1$

$\huge = \boxed{1}$