The limit represents the derivative of some function *f* at some number *a*. State such an *f* and *a*.

lim *x*→5 2**x* − 32 */ x* − 5

We are given the limit:

\[
\lim_{x \to 5} \frac{2^x - 32}{x - 5}
\]

This limit represents the derivative of some function \( f(x) \) at \( a = 5 \). To identify \( f(x) \) and \( a \), let's analyze the structure of the limit.

This is in the form of the definition of the derivative:

\[
f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}
\]

By comparing the given limit with this formula, we can deduce that:

- \( f(x) = 2^x \)
- \( a = 5 \)

This is because \( 32 = 2^5 \), so the expression \( 2^x - 32 \) can be rewritten as \( f(x) - f(5) \), where \( f(x) = 2^x \) and \( f(5) = 32 \).

Thus, the given limit represents the derivative of \( f(x) = 2^x \) at \( a = 5 \).

Would you like a more detailed explanation on the derivative or solving limits?

Here are five related questions:
1. What is the derivative of \( 2^x \)?
2. How does the limit definition of the derivative work?
3. What techniques can we use to evaluate limits like this?
4. Can the derivative of \( 2^x \) be generalized for any base \( b^x \)?
5. How can you apply the chain rule to exponential functions?

**Tip**: When solving limits, rewriting expressions to match known forms, like the derivative definition, can often simplify the problem.

The limit represents the derivative of some function f at some number a. State such an f and a. lim x→5 2^x − 32 / (x − 5)

This problem explores the limit representing the derivative of the exponential function 2^x at x = 5. Learn how to apply the definition of the derivative using limits and solve this advanced calculus problem. Suitable for Grades 11-12 students.

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