Use the limit definition of the derivative to find the slope of the tangent line to the curve                at the point . Then find an equation of the tangent line.

To find the slope of the tangent line to a curve at a given point using the limit definition of the derivative, we proceed step by step.

### Step 1: Recall the limit definition of the derivative
The derivative of a function \( f(x) \) at a point \( x = a \) is defined as:

\[
f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}
\]

This derivative represents the slope of the tangent line at the point \( (a, f(a)) \).

### Step 2: Define the function and the point
Let the function be \( f(x) \), and the point of interest be \( (a, f(a)) \).

- You would need to specify the equation of the curve (i.e., \( f(x) \)) and the point \( a \) at which we are finding the slope.
  
If you provide the curve equation (e.g., \( f(x) = x^2 \), or \( f(x) = \sin(x) \), etc.) and the point where the tangent line is to be calculated, I can proceed with the calculation.

### Step 3: Find the derivative using the limit definition
Once I have the function and the point, I will:

1. Substitute the function \( f(x) \) into the limit definition.
2. Simplify the expression.
3. Compute the limit as \( h \to 0 \), which will give the slope of the tangent line at \( x = a \).

### Step 4: Find the equation of the tangent line
The equation of the tangent line at \( x = a \) is:

\[
y - f(a) = f'(a)(x - a)
\]

Once the slope \( f'(a) \) is determined, this will give the equation of the tangent line.

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Could you please provide the function and the specific point so I can proceed with the solution?

Let me know if you'd like further clarification or details. Here are some questions to help you explore this topic further:

1. What is the significance of using the limit definition to find a derivative?
2. How does the tangent line relate to the graph of the function at a point?
3. Why does the slope of the tangent line represent the instantaneous rate of change?
4. What are alternative methods for finding the derivative besides the limit definition?
5. Can a tangent line ever intersect the curve at more than one point?

**Tip**: When using the limit definition, focus on simplifying the expression inside the limit carefully before attempting to take the limit as \( h \to 0 \).

Use the limit definition of the derivative to find the slope of the tangent line to the curve at the point. Then find an equation of the tangent line.

Learn how to find the slope of the tangent line to a curve at a point using the limit definition of the derivative. This detailed explanation covers step-by-step derivation and the equation of the tangent line, ideal for high school calculus students.

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