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What is the equation of the tangent line to f at a using definition of limit?

The expression in the image you provided is the derivative of a function \( f(x) \) at a point \( a \) using the definition of the derivative:

\[
f'(a) = \lim_{h \to 0} \frac{\frac{2}{1+h} - 2}{h}
\]

This appears to represent the derivative of the function \( f(x) = \frac{2}{x} \), evaluated at \( a = 1 \), based on the structure of the difference quotient. Let's verify this and then find the equation of the tangent line.

### Step-by-step Solution:

#### 1. Find the derivative \( f'(1) \)
We are given the limit expression:

\[
f'(1) = \lim_{h \to 0} \frac{\frac{2}{1+h} - 2}{h}
\]

Simplify the numerator:
\[
\frac{2}{1+h} - 2 = \frac{2 - 2(1+h)}{1+h} = \frac{2 - 2 - 2h}{1+h} = \frac{-2h}{1+h}
\]

Now, substitute this into the limit expression:
\[
f'(1) = \lim_{h \to 0} \frac{\frac{-2h}{1+h}}{h}
\]

Simplify further:
\[
f'(1) = \lim_{h \to 0} \frac{-2h}{h(1+h)} = \lim_{h \to 0} \frac{-2}{1+h}
\]

As \( h \to 0 \), this becomes:
\[
f'(1) = \frac{-2}{1+0} = -2
\]

#### 2. Equation of the tangent line at \( x = 1 \)
The equation of the tangent line at a point \( (a, f(a)) \) is given by the point-slope form:
\[
y - f(a) = f'(a)(x - a)
\]

We know that \( f(x) = \frac{2}{x} \), so:
\[
f(1) = \frac{2}{1} = 2
\]
And we already found that \( f'(1) = -2 \). Therefore, the equation of the tangent line is:
\[
y - 2 = -2(x - 1)
\]
Simplifying:
\[
y - 2 = -2x + 2
\]
\[
y = -2x + 4
\]

### Conclusion:
The equation of the tangent line to the curve \( f(x) = \frac{2}{x} \) at \( x = 1 \) is:
\[
y = -2x + 4
\]

Would you like more details on any part of the process?

### Follow-up Questions:
1. What is the geometric meaning of a derivative at a point?
2. How would the tangent line change for other values of \( a \)?
3. Can you find the second derivative of \( f(x) = \frac{2}{x} \)?
4. What does the derivative of \( f(x) \) tell you about the concavity of the function?
5. How does the point-slope form of a line relate to the derivative of a function?

### Tip:
When simplifying limits, always check if you can cancel common terms before directly evaluating the limit.

Step-by-step solution to finding the tangent line to the curve f(x) = 2/x at x = 1 using the definition of the derivative. This solution covers key calculus concepts such as limits, derivatives, and the point-slope form of a line. Suitable for Grades 10-12.

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