https://file.aiquickdraw.com/mathbot/file/1726009535624sk8cei15.jpg

To solve the problem from the image and find the slope of the tangent line, we will follow these steps:

### Part (a) - Find the Slope of the Tangent Line Using the Limit Definition

The slope of the tangent line, \( m_{\text{tan}} \), at a point \( P(a, f(a)) \) is given by the derivative of \( f(x) \) evaluated at \( a = 1 \):
\[
m_{\text{tan}} = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}.
\]

Given the function:
\[
f(x) = \frac{1}{4 + 3x},
\]
and the point \( P\left(1, \frac{1}{7}\right) \), we want to compute:
\[
m_{\text{tan}} = \lim_{h \to 0} \frac{f(1 + h) - f(1)}{h}.
\]

### Step 1: Compute \( f(1) \)

Substitute \( x = 1 \) into the function:
\[
f(1) = \frac{1}{4 + 3(1)} = \frac{1}{7}.
\]

### Step 2: Compute \( f(1+h) \)

Substitute \( 1 + h \) for \( x \) in the function:
\[
f(1 + h) = \frac{1}{4 + 3(1 + h)} = \frac{1}{7 + 3h}.
\]

### Step 3: Set up the Difference Quotient

Now, substitute \( f(1+h) \) and \( f(1) \) into the difference quotient:
\[
m_{\text{tan}} = \lim_{h \to 0} \frac{\frac{1}{7 + 3h} - \frac{1}{7}}{h}.
\]

### Step 4: Simplify the Difference Quotient

We need to simplify the expression \( \frac{1}{7 + 3h} - \frac{1}{7} \) by finding a common denominator:
\[
\frac{1}{7 + 3h} - \frac{1}{7} = \frac{7 - (7 + 3h)}{7(7 + 3h)} = \frac{-3h}{7(7 + 3h)}.
\]

Thus, the difference quotient becomes:
\[
m_{\text{tan}} = \lim_{h \to 0} \frac{\frac{-3h}{7(7 + 3h)}}{h} = \lim_{h \to 0} \frac{-3}{7(7 + 3h)}.
\]

### Step 5: Take the Limit as \( h \to 0 \)

Now, evaluate the limit as \( h \to 0 \):
\[
m_{\text{tan}} = \frac{-3}{7(7 + 0)} = \frac{-3}{49}.
\]

### Final Answer:

The slope of the tangent line is:
\[
m_{\text{tan}} = \frac{-3}{49}.
\]

### Part (b) - Equation of the Tangent Line

The equation of the tangent line at point \( P(1, \frac{1}{7}) \) can be written using the point-slope form of a line:
\[
y - y_1 = m(x - x_1),
\]
where \( m = \frac{-3}{49} \), \( x_1 = 1 \), and \( y_1 = \frac{1}{7} \).

Substitute the values:
\[
y - \frac{1}{7} = \frac{-3}{49}(x - 1).
\]

Simplify:
\[
y = \frac{-3}{49}(x - 1) + \frac{1}{7}.
\]

If you'd like, I can expand and simplify this further.

Would you like a more detailed expansion of the tangent line equation, or do you have any questions?

### Relative Questions:
1. How do you compute the derivative using limits?
2. What is the geometric interpretation of a tangent line to a curve?
3. How do you find the slope of a curve at any given point?
4. Can you explain the difference between the secant line and the tangent line?
5. What is the role of limits in calculus, particularly in finding slopes of curves?

### Tip:
Always simplify the function inside the limit before taking the limit—it can significantly reduce the complexity of the computation.

Use the definition of the slope of the tangent line, m_{tan} = lim (h -> 0) (f(a+h) - f(a)) / h, to find the slope of the tangent line to the graph of f at P. Determine the equation of the tangent line at P. Given f(x) = 1 / (4 + 3x), and the point P(1, 1/7).

Learn how to find the slope of the tangent line to the curve f(x) = 1 / (4 + 3x) at the point P(1, 1/7) using the limit definition of a derivative. Includes step-by-step explanation for students in Grades 11-12.

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