límite cuando x tiende a 0+de x^2*e^{1/x}

Solução

x \to 0 + lim (x^{2} \cdot e^{\frac{1}{x}})

\infty

Passos da solução

x \to 0 + lim (x^{2} e^{\frac{1}{x}})

Reescreva seguindo a regra de L'Hopital

= x \to 0 + lim (\frac{e ^{\frac{1}{x}}}{\frac{1}{x ^{2}}})

Aplicar a regra de L'Hopital

= x \to 0 + lim \frac{- \frac{e ^{\frac{1}{x}}}{x ^{2}}}{- \frac{2}{x ^{3}}}

Simplificar

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

= x \to 0 + lim (\frac{1}{2} e^{\frac{1}{x}} x)

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

= \frac{1}{2} \cdot x \to 0 + lim (e^{\frac{1}{x}} x)

Reescreva seguindo a regra de L'Hopital

= \frac{1}{2} \cdot x \to 0 + lim (\frac{e ^{\frac{1}{x}}}{\frac{1}{x}})

Aplicar a regra de L'Hopital

= \frac{1}{2} \cdot x \to 0 + lim \frac{- \frac{e ^{\frac{1}{x}}}{x ^{2}}}{- \frac{1}{x ^{2}}}

Simplificar

\frac{- \frac{e ^{\frac{1}{x}}}{x ^{2}}}{- \frac{1}{x ^{2}}} : e^{\frac{1}{x}}

= \frac{1}{2} \cdot x \to 0 + lim (e^{\frac{1}{x}})

Aplicar a regra da cadeia:

\infty

= \frac{1}{2} \cdot \infty

Aplicar as propriedades dos limites infinitos:

c \cdot \infty = \infty

= \infty

x \to 0 lim (x^{2} - x + 1)

t \to 0 + lim (t ln (t))

x \to 2 lim (\frac{x - 3}{x + 1})

x \to - 2 lim (F (x))

x \to - 1 lim (x^{12})