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

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

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

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

\infty

Solution steps

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

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

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

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

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

= \frac{\infty}{1}

Apply Infinity Property:

\frac{\infty}{c} = \infty

= \infty

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