What is the limit as x approaches 0+of x^2*e^{1/x} ?

The limit as x approaches 0+of x^2*e^{1/x} is infinity

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limit as x approaches 0+of x^2*e^{1/x}

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

\infty

Solution steps

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

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

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

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

Simplify

\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)

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

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

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

Simplify

\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}})

Apply the Limit Chain Rule:

\infty

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

Apply Infinity Property:

c \cdot \infty = \infty

= \infty

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

x \to 1 lim (csc (π x) ln (x))

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

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

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

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