Let's simplify and evaluate the limit:

We are given:

$\lim_{x \to 0^+} \frac{\frac{1}{x}}{-\frac{1}{2}x^{-3/2}}$

Step 1: Rewrite the expression

Simplify the denominator first:

$-\frac{1}{2}x^{-3/2} = -\frac{1}{2} \cdot \frac{1}{x^{3/2}} = -\frac{1}{2x^{3/2}}$

Thus, the expression becomes:

$\frac{\frac{1}{x}}{-\frac{1}{2x^{3/2}}}$

Step 2: Simplify the fraction

To divide by a fraction, multiply by its reciprocal:

$\frac{\frac{1}{x}}{-\frac{1}{2x^{3/2}}} = \frac{1}{x} \cdot \frac{-2x^{3/2}}{1}$

Simplify the multiplication:

$= -2 \cdot \frac{x^{3/2}}{x}$

Combine the powers of $x$ ($x^{3/2}/x = x^{1/2}$):

$= -2x^{1/2}$

Step 3: Evaluate the limit as $x \to 0^+$

As $x \to 0^+$, $x^{1/2} \to 0$. Hence:

$\lim_{x \to 0^+} -2x^{1/2} = 0$

Final Answer:

$\boxed{0}$

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Would you like a breakdown of any specific step or more examples of similar problems?

5 Related Questions:

1. How do you simplify expressions involving powers of $x$?
2. What is the general rule for dividing two functions with exponents?
3. How does the behavior of $x^{1/2}$ affect limits as $x \to 0^+$?
4. What is the interpretation of limits involving infinity or zero?
5. How do you handle limits involving reciprocal functions?

Tip:

When simplifying fractions with exponents, remember to use the rules $x^a / x^b = x^{a-b}$.