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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 \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \).

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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.

Each limit represents the derivative of some function f at some number a. State such an f and a in each case.

Solve limits that represent the derivatives of functions like square root, exponential, polynomial, tangent, and sine at specific points. This problem covers key differential calculus concepts, including the formal definition of a derivative and important theorems.

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