The limit represents f ′(c) for a function f(x) and a number c. Find f(x) and c.
The limit as x approaches 4 when f(x) = -x^2 + 16)/(x − 4)

We are tasked with evaluating the limit:

\[
\lim_{x \to 4} \frac{f(x) = -x^2 + 16}{x - 4}
\]

This expression seems to have the indeterminate form \( \frac{0}{0} \) when we substitute \(x = 4\) directly. To resolve this, we need to simplify the expression.

### Step 1: Simplify the numerator

The numerator is \( -x^2 + 16 \), which can be rewritten as:

\[
-x^2 + 16 = -(x^2 - 16) = -(x - 4)(x + 4)
\]

Thus, the limit becomes:

\[
\lim_{x \to 4} \frac{-(x - 4)(x + 4)}{x - 4}
\]

### Step 2: Cancel the common factor

The \( (x - 4) \) terms cancel out (as long as \( x \neq 4 \)), leaving us with:

\[
\lim_{x \to 4} -(x + 4)
\]

### Step 3: Evaluate the limit

Now, substitute \( x = 4 \) into the simplified expression:

\[
-(4 + 4) = -8
\]

Thus, the limit is:

\[
\boxed{-8}
\]

### Identifying \( f(x) \) and \( c \)

We are also asked to identify \( f(x) \) and \( c \):

- The function \( f(x) \) is likely the original function \( f(x) = -x^2 + 16 \).
- The number \( c \) is the value at which we are taking the limit, \( c = 4 \).

Would you like more details or have any questions? Here are 5 related questions:

1. How would the limit change if \(f(x)\) was a different quadratic function?
2. What happens if we approach a different value of \(c\) instead of 4?
3. How can we apply L'Hopital's Rule to evaluate limits like this?
4. What if \(f(x)\) involved a higher degree polynomial, like \(x^3\)?
5. How do limits relate to the concept of a derivative in calculus?

**Tip:** Always check for indeterminate forms like \( \frac{0}{0} \) before simplifying limits involving polynomials.

The limit represents f ′(c) for a function f(x) and a number c. Find f(x) and c. The limit as x approaches 4 when f(x) = (-x^2 + 16)/(x − 4).

Learn how to evaluate the limit of the function (-x^2 + 16)/(x - 4) as x approaches 4. This solution covers key concepts in algebra and calculus, including indeterminate forms and factorization. Perfect for students in Grades 11-12 studying limits and continuity.

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