Limits of the form lim as h approaches o of the function (f(x+h) - f(x))/h occur frequently in calculus. evalute this limit for the given value of x and function f. F(x) = x^2 and x = -1

The limit you are referring to is the definition of the **derivative** of a function \( f(x) \) at a point \( x \). In this case, we are tasked with evaluating the derivative of \( f(x) = x^2 \) at \( x = -1 \).

The general form of the derivative definition is:

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

### Step-by-Step Solution:

1. **Start with the function:**
   \[
   f(x) = x^2
   \]

2. **Apply the limit definition of the derivative:**
   \[
   f'(x) = \lim_{h \to 0} \frac{(f(x+h) - f(x))}{h}
   \]

3. **Substitute the given function \( f(x) = x^2 \) into the expression:**
   \[
   f'(x) = \lim_{h \to 0} \frac{((x+h)^2 - x^2)}{h}
   \]

4. **Expand \( (x+h)^2 \):**
   \[
   (x+h)^2 = x^2 + 2xh + h^2
   \]
   So the expression becomes:
   \[
   f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h}
   \]

5. **Simplify by canceling terms:**
   \[
   f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h}
   \]
   Factor \( h \) out of the numerator:
   \[
   f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h}
   \]

6. **Cancel \( h \) from the numerator and denominator:**
   \[
   f'(x) = \lim_{h \to 0} (2x + h)
   \]

7. **Evaluate the limit as \( h \to 0 \):**
   \[
   f'(x) = 2x
   \]

8. **Finally, substitute \( x = -1 \):**
   \[
   f'(-1) = 2(-1) = -2
   \]

### Final Answer:
The derivative of \( f(x) = x^2 \) at \( x = -1 \) is \( f'(-1) = -2 \).

---

Would you like further details on this process or any part of the derivative definition? 

Here are 5 additional questions for deeper understanding:
1. How is the derivative related to the slope of the tangent line?
2. What does the derivative represent geometrically?
3. How would you differentiate \( f(x) = x^3 \) using the same limit definition?
4. What happens if you evaluate the derivative at \( x = 1 \) for \( f(x) = x^2 \)?
5. How can the power rule be derived from the limit definition of a derivative?

**Tip**: Always expand and simplify algebraic expressions when working with limits involving derivatives!

Limits of the form lim as h approaches 0 of the function (f(x+h) - f(x))/h occur frequently in calculus. Evaluate this limit for the given value of x and function f. F(x) = x^2 and x = -1.

Learn how to evaluate the derivative of f(x) = x^2 at x = -1 using the limit definition. This problem walks through the steps of applying the limit formula, expanding algebraic expressions, and simplifying to find f'(-1) = -2. Suitable for high school and early college students studying calculus.

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