What is the limit as x approaches infinity of ((e^{-xy-x+y}))/(x^2y^2+xy) ?

The limit as x approaches infinity of ((e^{-xy-x+y}))/(x^2y^2+xy) is 0

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limit as x approaches infinity of ((e^{-xy-x+y}))/(x^2y^2+xy)

x \to \infty lim (\frac{( e ^{- x y - x + y} )}{x ^{2} y ^{2} + x y})

0

Solution steps

x \to \infty lim (\frac{e ^{- x y - x + y}}{x ^{2} y ^{2} + x y})

Simplify

e^{- x y - x + y} : e^{- x y} e^{- x} e^{y}

= x \to \infty lim (\frac{e ^{- x y} e ^{- x} e ^{y}}{x ^{2} y ^{2} + x y})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= e^{y} \cdot x \to \infty lim (\frac{e ^{- x y} e ^{- x}}{x ^{2} y ^{2} + x y})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

= e^{y} \frac{lim _{x \to \infty} ( e ^{- x y} e ^{- x} )}{lim _{x \to \infty} ( x ^{2} y ^{2} + x y )}

x \to \infty lim (e^{- x y} e^{- x}) = 0

x \to \infty lim (x^{2} y^{2} + x y) = \infty

= e^{y} \frac{0}{\infty}

Simplify

e^{y} \frac{0}{\infty} : 0

= 0

y = x^{2} - 3, y = x + 3

\int \frac{4 x - 11}{x ^{3} - 9 x ^{2}} d x

y^{^{''}} - 2 y^{^{'}} + 8 y = 320 t^{2}

\frac{\partial}{\partial z} (- 2 x z)

\frac{1}{3 x}

What is the limit as x approaches infinity of ((e^{-xy-x+y}))/(x^2y^2+xy) ?
The limit as x approaches infinity of ((e^{-xy-x+y}))/(x^2y^2+xy) is 0