Question

The value of the following limit is \(\underline{\hspace{1cm}}\). \[ \lim_{x \to 0^+} \frac{\sqrt{x}}{1 - e^{2\sqrt{x}}} \]

Hint:

For small \( x \), use approximations such as \( e^{2\sqrt{x}} \approx 1 + 2\sqrt{x} \) to simplify limits involving exponential functions.

Answer 1

Correct Answer: -0.5

To solve the limit, we can use the approximation for \( e^y \) for small \( y \), where \( e^y \approx 1 + y \) for \( y \to 0 \). In this case, \( y = 2\sqrt{x} \), and as \( x \to 0^+ \), \( 2\sqrt{x} \) approaches 0. Thus: \[ e^{2\sqrt{x}} \approx 1 + 2\sqrt{x}. \] Substituting this approximation into the limit: \[ \frac{\sqrt{x}}{1 - e^{2\sqrt{x}}} \approx \frac{\sqrt{x}}{1 - (1 + 2\sqrt{x})} = \frac{\sqrt{x}}{-2\sqrt{x}} = -\frac{1}{2}. \] Thus, the value of the limit is: \[ \boxed{-0.5}. \]