mathop {lim }limits_{x to 0^ + } frac{x e^{1/ | vedclass

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Question

(A) To evaluate the limit $\mathop {\lim }\limits_{x 	o 0^ + } \frac{x e^{1/x}}{1 + e^{1/x}}$,we divide the numerator and the denominator by $e^{1/x}

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(A) To evaluate the limit $\mathop {\lim }\limits_{x 	o 0^ + } \frac{x e^{1/x}}{1 + e^{1/x}}$,we divide the numerator and the denominator by $e^{1/x}$:$\mathop {\lim }\limits_{x 	o 0^ + } \frac{x}{\frac{1}{e^{1/x}} + 1} = \mathop {\lim }\limits_{x 	o 0^ + } \frac{x}{e^{-1/x} + 1}$As $x 	o 0^ +$,the term $1/x 	o \infty$,so $e^{-1/x} 	o 0$.Substituting these values,we get $\frac{0}{0 + 1} = 0$.

$\mathop {\lim }\limits_{x \to 0^ + } \frac{x e^{1/x}}{1 + e^{1/x}} = $

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