This is a limit problem involving radicals. The expression can be simplified using the technique of multiplying by the conjugate to eliminate the square roots.

The limit is:

$\lim_{{h \to 0}} \frac{\sqrt{11 + h} - \sqrt{11}}{h}$

Step 1: Multiply by the conjugate

To simplify, we multiply the numerator and denominator by the conjugate of the numerator, which is $\sqrt{11 + h} + \sqrt{11}$:

$\frac{\sqrt{11 + h} - \sqrt{11}}{h} \cdot \frac{\sqrt{11 + h} + \sqrt{11}}{\sqrt{11 + h} + \sqrt{11}} = \frac{\left(\sqrt{11 + h}\right)^2 - \left(\sqrt{11}\right)^2}{h \left(\sqrt{11 + h} + \sqrt{11}\right)}$

Step 2: Simplify the numerator

Using the difference of squares formula:

$\left(\sqrt{11 + h}\right)^2 - \left(\sqrt{11}\right)^2 = (11 + h) - 11 = h$

Thus, the expression becomes:

$\frac{h}{h \left(\sqrt{11 + h} + \sqrt{11}\right)}$

Step 3: Cancel out the $h$

We can cancel the $h$ in the numerator and denominator:

$\frac{1}{\sqrt{11 + h} + \sqrt{11}}$

Step 4: Take the limit as $h \to 0$

Now, substitute $h = 0$ into the expression:

$\frac{1}{\sqrt{11 + 0} + \sqrt{11}} = \frac{1}{2\sqrt{11}}$

So, the final result is:

$\lim_{{h \to 0}} \frac{\sqrt{11 + h} - \sqrt{11}}{h} = \frac{1}{2\sqrt{11}}$

Would you like further details on any of these steps or have questions?

Related Questions:

1. What is the significance of multiplying by the conjugate in limits?
2. Can we solve similar limits for other values, like $\sqrt{a + h} - \sqrt{a}$?
3. How do we deal with indeterminate forms in limit problems?
4. What are other common techniques for simplifying limits involving square roots?
5. How can the limit definition of the derivative be applied here?

Tip: Always look for indeterminate forms like $\frac{0}{0}$ when solving limit problems, as they often signal the need for algebraic simplifications.