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

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

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

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

0

Solution steps

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

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

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

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

Simplify

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

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

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

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

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

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

x \to 0 + lim (1) = 1

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

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

Simplify

2 \cdot \frac{1}{\infty} : 0

= 0

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x \to 3 lim (x - 7)

x \to 0 lim ((\frac{1 + 3 ^{x}}{2})^{\frac{1}{x}})

z \to i lim (\frac{z ^{4} - 1}{z - i})

x \to - \infty lim (\frac{x}{1})

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