https://file.aiquickdraw.com/mathbot/file/1727655281026yhr32pe8.jpg

Let's solve Problem #10 based on the image provided.

### Given function:
\[ f(x) = \frac{7}{x} \]

#### Part (a): 
We need to find which of the given limits represents the slope of the tangent line to the graph of \( f(x) \) at \( x = 3 \). The slope of the tangent line is the derivative \( f'(x) \) evaluated at \( x = 3 \).

The derivative of a function at a point can be expressed using the limit definition of the derivative:
\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

Substituting \( f(x) = \frac{7}{x} \) and \( x = 3 \), we calculate:
\[
f'(3) = \lim_{h \to 0} \frac{\frac{7}{3+h} - \frac{7}{3}}{h}
\]

We now look for this expression among the options in the image. The matching expression is **Option (C)**:
\[
\lim_{h \to 0} \frac{\frac{7}{3+h} - \frac{7}{3}}{h}
\]
Thus, the correct answer for part (a) is **Option C**.

#### Part (b):
Now, we need to evaluate this limit and use the result to find the equation of the tangent line at \( x = 3 \).

Let's evaluate the limit:
\[
f'(3) = \lim_{h \to 0} \frac{\frac{7}{3+h} - \frac{7}{3}}{h}
\]
1. First, find a common denominator for the expression inside the limit:
\[
\frac{7}{3+h} - \frac{7}{3} = \frac{7(3) - 7(3+h)}{3(3+h)} = \frac{21 - 21 - 7h}{3(3+h)} = \frac{-7h}{3(3+h)}
\]
2. Simplifying the limit:
\[
f'(3) = \lim_{h \to 0} \frac{\frac{-7h}{3(3+h)}}{h} = \lim_{h \to 0} \frac{-7h}{h \cdot 3(3+h)} = \lim_{h \to 0} \frac{-7}{3(3+h)}
\]
3. As \( h \to 0 \), this becomes:
\[
f'(3) = \frac{-7}{9}
\]

Thus, the slope of the tangent line at \( x = 3 \) is \( m = \frac{-7}{9} \).

Next, we find the equation of the tangent line. The general form of the equation of a tangent line is:
\[
y - f(x_0) = m(x - x_0)
\]
where \( m = \frac{-7}{9} \), and \( x_0 = 3 \). We also need \( f(3) \):
\[
f(3) = \frac{7}{3}
\]

Now, using the point \( (3, \frac{7}{3}) \) and slope \( m = \frac{-7}{9} \):
\[
y - \frac{7}{3} = \frac{-7}{9}(x - 3)
\]
Simplifying this equation:
\[
y = \frac{-7}{9}(x - 3) + \frac{7}{3}
\]
Distribute the slope:
\[
y = \frac{-7}{9}x + \frac{7}{3} + \frac{7}{3} = \frac{-7}{9}x + \frac{14}{3}
\]

Thus, the equation of the tangent line is:
\[
y = \frac{-7}{9}x + \frac{14}{3}
\]

---

Would you like to explore any of these steps further or need additional details?

Here are 5 related questions:
1. How do we compute the derivative of a function using the limit definition?
2. How do you find the equation of a tangent line for any function at a given point?
3. Why does the limit process give us the slope of the tangent line?
4. What is the geometric interpretation of a derivative at a specific point?
5. Can this method be applied to other types of functions, such as quadratic or exponential functions?

**Tip**: When using the limit definition of a derivative, always simplify the difference between function values before dividing by \( h \) to make sure the \( h \) in the denominator cancels properly.

Let f(x) = 7/x. (a) Which of the given limits is equal to the slope of the tangent line to the graph of f(x) at x = 3? (b) Evaluate the limit in (a) and then use the result to find the equation of the tangent line to the graph of f(x) at x = 3.

Learn how to find the slope of the tangent line to the graph of f(x) = 7/x at x = 3 using the limit definition of the derivative. Step-by-step solution for evaluating the limit and finding the equation of the tangent line.

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