Evaluate `lim_(hto0) [(1)/(h^(3)sqrt(8+h))-(1)/(2h)].`

Explanation

To evaluate the limit

\[ \lim_{h \to 0} \left( \frac{1}{h^3 \sqrt{8+h}} - \frac{1}{2h} \right), \]

we will follow these steps:

### Step 1: Combine the fractions
First, we will find a common denominator for the two fractions. The common denominator is \(2h^3 \sqrt{8+h}\).

\[ \frac{1}{h^3 \sqrt{8+h}} - \frac{1}{2h} = \frac{2 - h^2 \sqrt{8+h}}{2h^3 \sqrt{8+h}}. \]

### Step 2: Factor out \(h\)
Now we can factor out \(h\) from the numerator:

\[ \frac{2 - h^2 \sqrt{8+h}}{2h^3 \sqrt{8+h}} = \frac{-h^2 \sqrt{8+h} + 2}{2h^3 \sqrt{8+h}}. \]

### Step 3: Rewrite the limit
We can rewrite the limit as:

\[ \lim_{h \to 0} \frac{-h^2 \sqrt{8+h} + 2}{2h^3 \sqrt{8+h}}. \]

### Step 4: Simplify the limit
As \(h \to 0\), \(\sqrt{8+h} \to \sqrt{8} = 2\sqrt{2}\). Thus, we can substitute this into our limit:

\[ \lim_{h \to 0} \frac{-h^2 (2\sqrt{2}) + 2}{2h^3 (2\sqrt{2})}. \]

### Step 5: Evaluate the limit
Now we can evaluate the limit:

\[ \lim_{h \to 0} \frac{-2\sqrt{2}h^2 + 2}{4\sqrt{2}h^3} = \lim_{h \to 0} \frac{2(1 - \sqrt{2}h^2)}{4\sqrt{2}h^3}. \]

As \(h\) approaches 0, the term \(-\sqrt{2}h^2\) approaches 0, so we have:

\[ \lim_{h \to 0} \frac{2}{4\sqrt{2}h^3} = \lim_{h \to 0} \frac{1}{2\sqrt{2}h^3}. \]

This limit diverges to \(-\infty\) as \(h\) approaches 0.

### Final Answer
Thus, the limit is:

\[ \lim_{h \to 0} \left( \frac{1}{h^3 \sqrt{8+h}} - \frac{1}{2h} \right) = -\infty. \]