Suppose 
f(x) = 

x + 5
.
 Evaluate the limit below by using algebra to simplify the difference quotient (in the first answer blank) and then evaluating the limit (in the second answer blank).
lim 
h→0 

f(2 + h) − f(2)
h

 = 
lim 
h→0 

To evaluate the limit, we first need to simplify the difference quotient. Let's go step by step.

### Step 1: Express the Difference Quotient

Given \( f(x) = x + 5 \), we want to evaluate:

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

### Step 2: Calculate \( f(2 + h) \) and \( f(2) \)

- \( f(2 + h) = (2 + h) + 5 = 2 + h + 5 = h + 7 \)
- \( f(2) = 2 + 5 = 7 \)

### Step 3: Substitute into the Difference Quotient

Now substitute \( f(2 + h) \) and \( f(2) \) into the difference quotient:

\[
\frac{f(2 + h) - f(2)}{h} = \frac{(h + 7) - 7}{h} = \frac{h}{h}
\]

### Step 4: Simplify the Expression

Simplify the difference quotient:

\[
\frac{h}{h} = 1
\]

### Step 5: Evaluate the Limit

Now, evaluate the limit as \( h \) approaches 0:

\[
\lim_{h \to 0} 1 = 1
\]

### Final Answer

So the limit is:

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

---

Would you like further details or have any questions?

Here are some related questions:
1. How do you calculate the derivative using the difference quotient?
2. Can the limit process be used to find the derivative of any function?
3. What happens if \( f(x) \) is a more complex function, like a quadratic or cubic function?
4. How is the limit definition of the derivative related to the tangent line of a function?
5. What are the potential pitfalls when evaluating limits involving difference quotients?

**Tip:** When dealing with difference quotients, always simplify the algebraic expression before evaluating the limit to avoid indeterminate forms like \(\frac{0}{0}\).

Suppose f(x) = x + 5. Evaluate the limit below by using algebra to simplify the difference quotient and then evaluating the limit: lim (h→0) [f(2 + h) − f(2)] / h.

Discover how to evaluate the limit of the difference quotient for the linear function f(x) = x + 5. This problem involves simplifying the difference quotient and applying limit concepts, suitable for high school students in Grades 10-12.

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