The image you provided contains several limits that represent the derivatives of some function $f(x)$ at a specific point $a$. Let's solve them one by one.

3. $\lim_{h \to 0} \frac{\sqrt{9+h} - 3}{h}$

This limit is of the form of the derivative definition: $\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$ It matches with the derivative of the function $f(x) = \sqrt{x}$ at $a = 9$.

4. $\lim_{h \to 0} \frac{e^{-2+h} - e^{-2}}{h}$

This limit represents the derivative of $f(x) = e^x$ at $a = -2$, because it's of the form: $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ with $f(x) = e^x$ and $a = -2$.

5. $\lim_{x \to 2} \frac{x^6 - 64}{x - 2}$

This limit represents the derivative of the function $f(x) = x^6$ at $a = 2$, because: $f'(x) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ In this case, $f(2) = 64$, so the function is $f(x) = x^6$ and $a = 2$.

6. $\lim_{x \to 1/4} \frac{\frac{1}{x} - 4}{x - 1/4}$

This limit represents the derivative of $f(x) = \frac{1}{x}$ at $a = \frac{1}{4}$.

7. $\lim_{h \to 0} \frac{\tan\left(\frac{\pi}{4} + h\right) - 1}{h}$

This limit represents the derivative of $f(x) = \tan(x)$ at $a = \frac{\pi}{4}$, because the derivative of $\tan(x)$ is defined as: $\lim_{h \to 0} \frac{\tan(a + h) - \tan(a)}{h}$ with $\tan\left(\frac{\pi}{4}\right) = 1$.

8. $\lim_{\theta \to \pi/6} \frac{\sin(\theta) - \frac{1}{2}}{\theta - \frac{\pi}{6}}$

This is the derivative of $f(\theta) = \sin(\theta)$ at $a = \frac{\pi}{6}$, because: $f'(a) = \lim_{\theta \to a} \frac{\sin(\theta) - \sin(a)}{\theta - a}$ with $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$.

Let me know if you'd like further details or clarifications. Here are 5 related questions to consider:

1. How do you differentiate a function using the formal definition of a derivative?
2. Can you identify other common functions where the derivative at specific points is useful?
3. What is the geometric interpretation of a derivative at a point?
4. How does the chain rule apply when differentiating composite functions?
5. Can you explore the concept of higher-order derivatives from these examples?

Tip: Always remember that limits involving the difference quotient are essential in finding derivatives, especially when the direct formula for the derivative isn't immediately clear.