To find the value of $\frac{f(x+h) - f(x)}{h}$ for the function $f(x) = x^2 + 7$, let's follow these steps:

1. Calculate $f(x + h)$:

   Given $f(x) = x^2 + 7$, we substitute $x + h$ into the function: $f(x + h) = (x + h)^2 + 7$

2. Expand $f(x + h)$:

   $f(x + h) = (x + h)^2 + 7 = x^2 + 2xh + h^2 + 7$

3. Find $f(x + h) - f(x)$:

   We have: $f(x + h) - f(x) = (x^2 + 2xh + h^2 + 7) - (x^2 + 7)$ Simplify the expression: $f(x + h) - f(x) = x^2 + 2xh + h^2 + 7 - x^2 - 7 = 2xh + h^2$

4. Divide by $h$:

   $\frac{f(x + h) - f(x)}{h} = \frac{2xh + h^2}{h}$ Simplify the fraction: $\frac{f(x + h) - f(x)}{h} = 2x + h$

So, the value of $\frac{f(x + h) - f(x)}{h}$ is $2x + h$.

Do you want details or have any questions?

Here are some related questions to further your understanding:

1. What is the significance of the expression $\frac{f(x + h) - f(x)}{h}$ in calculus?
2. How does this expression relate to the derivative of a function?
3. What would be the result if $h$ approaches 0 in $\frac{f(x + h) - f(x)}{h}$?
4. Can you derive the derivative of $f(x) = x^2 + 7$ using the limit definition?
5. How do you interpret the term $2x + h$ geometrically?
6. What happens to the expression $\frac{f(x + h) - f(x)}{h}$ if the function $f(x)$ is linear?
7. How would you find $\frac{f(x + h) - f(x)}{h}$ for a trigonometric function like $f(x) = \sin x$?
8. Can you explain the difference between a secant line and a tangent line in the context of this expression?

Tip: When working with limits, especially in calculus, always remember to consider the behavior of the function as the variable approaches the limit point. This approach is crucial for understanding derivatives and integrals.