Chapter 4 - Section 4.4 - Indeterminate Forms and l''Hospital''s Rule - 4.4 Exercises - Page 316: 19

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$$\eqalign{ & \mathop {\lim }\limits_{x \to \infty } \frac{{\sqrt x }}{{1 + {e^x}}} \cr & {\text{Use the quotient property of limits}} \cr & = \frac{{\mathop {\lim }\limits_{x \to \infty } \left[ {\sqrt x } \right]}}{{\mathop {\lim }\limits_{x \to \infty } \left( {1 + {e^x}} \right)}} \cr & {\text{Evaluate the limit when }}x \to \infty \cr & \frac{{\mathop {\lim }\limits_{x \to \infty } \left[ {\sqrt x } \right]}}{{\mathop {\lim }\limits_{x \to \infty } \left( {1 + {e^x}} \right)}} = \frac{\infty }{\infty } \cr & {\text{The limit is an indeterminate form type }}\frac{\infty }{\infty },{\text{ so we can apply}} \cr & {\text{the L'Hospital's Rule}}{\text{.}} \cr & \mathop {\lim }\limits_{x \to \infty } \frac{{\sqrt x }}{{1 + {e^x}}} = \mathop {\lim }\limits_{x \to \infty } \frac{{\frac{d}{{dx}}\left[ {\sqrt x } \right]}}{{\frac{d}{{dx}}\left[ {1 + {e^x}} \right]}} \cr & = \mathop {\lim }\limits_{x \to \infty } \frac{{\frac{1}{{2\sqrt x }}}}{{{e^x}}} \cr & = \mathop {\lim }\limits_{x \to \infty } \frac{1}{{2{e^x}\sqrt x }} \cr & = \frac{1}{{\mathop {\lim }\limits_{x \to \infty } \left( {2{e^x}\sqrt x } \right)}} \cr & {\text{Evaluate the limit when }}x \to \infty \cr & = 0 \cr & {\text{Therefore}}{\text{,}} \cr & \mathop {\lim }\limits_{x \to \infty } \frac{{\sqrt x }}{{1 + {e^x}}} = 0 \cr} $$