What is the limit as x approaches infinity of ((x+1)^2e^{x^3})/(e^{(x+1)^3)x^2} ?

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limit as x approaches infinity of ((x+1)^2e^{x^3})/(e^{(x+1)^3)x^2}

x \to \infty lim (\frac{( x + 1 ) ^{2} e ^{x^{3}}}{e ^{(x + 1)^{3}} x ^{2}})

0

Solution steps

x \to \infty lim (\frac{( x + 1 ) ^{2} e ^{x^{3}}}{e ^{(x + 1)^{3}} x ^{2}})

Simplify

\frac{( x + 1 ) ^{2} e ^{x^{3}}}{e ^{(x + 1)^{3}} x ^{2}} : \frac{e ^{- 3 x^{2} - 3 x - 1} ( x + 1 ) ^{2}}{x ^{2}}

= x \to \infty lim (\frac{e ^{- 3 x^{2} - 3 x - 1} ( x + 1 ) ^{2}}{x ^{2}})

Apply the following algebraic property

: a + b = a (1 + \frac{b}{a})

- 3 x^{2} - 3 x - 1 = x^{2} (- 3 - \frac{3}{x} - \frac{1}{x ^{2}})

= x \to \infty lim (\frac{e ^{x^{2} (- 3 - \frac{3}{x} - \frac{1}{x ^{2}})} ( x + 1 ) ^{2}}{x ^{2}})

Expand

\frac{e ^{x^{2} (- 3 - \frac{3}{x} - \frac{1}{x ^{2}})} ( x + 1 ) ^{2}}{x ^{2}} : e^{- 3 x^{2} - 3 x - 1} + \frac{2 e ^{- 3 x^{2} - 3 x - 1}}{x} + \frac{e ^{- 3 x^{2} - 3 x - 1}}{x ^{2}}

= x \to \infty lim (e^{- 3 x^{2} - 3 x - 1} + \frac{2 e ^{- 3 x^{2} - 3 x - 1}}{x} + \frac{e ^{- 3 x^{2} - 3 x - 1}}{x ^{2}})

x \to a lim [f (x) \pm g (x)] = x \to a lim f (x) \pm x \to a lim g (x)

With the exception of indeterminate form

= x \to \infty lim (e^{- 3 x^{2} - 3 x - 1}) + x \to \infty lim (\frac{2 e ^{- 3 x^{2} - 3 x - 1}}{x}) + x \to \infty lim (\frac{e ^{- 3 x^{2} - 3 x - 1}}{x ^{2}})

x \to \infty lim (e^{- 3 x^{2} - 3 x - 1}) = 0

x \to \infty lim (\frac{2 e ^{- 3 x^{2} - 3 x - 1}}{x}) = 0

x \to \infty lim (\frac{e ^{- 3 x^{2} - 3 x - 1}}{x ^{2}}) = 0

= 0 + 0 + 0

Simplify

= 0

x \to 16 - lim (\frac{8}{11 + 2 ^{\frac{1}{x - 16}}})

\int 2 (x^{2} + 2 x)^{16} (2 x + 2) d x

\frac{2 s}{( s ^{2} + 2 s + 6 )}

f (x) = 3^{x^{2} - 1}, at x = 2

y^{^{''}} - y = 4 sin (3 t)

What is the limit as x approaches infinity of ((x+1)^2e^{x^3})/(e^{(x+1)^3)x^2} ?
The limit as x approaches infinity of ((x+1)^2e^{x^3})/(e^{(x+1)^3)x^2} is 0