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Answer the questions in the picture briefly

We are given the function \( f(x) = ax + b \) and the limit equation:

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

We are tasked with finding \( f(1) \).

### Step 1: Find the value of \( a \)
We know that \( f(x) = ax + b \), so substitute it into the given limit equation:

\[
\lim_{{x \to 4}} \frac{ax + b}{\sqrt{x} - 2} = -4
\]

Since \( \lim_{{x \to 4}} \sqrt{x} = 2 \), we apply L'Hopital's Rule (because the direct substitution leads to an indeterminate form \( \frac{0}{0} \)):

Differentiate the numerator and denominator:

- The derivative of the numerator \( f(x) = ax + b \) is \( f'(x) = a \).
- The derivative of the denominator \( \sqrt{x} - 2 \) is \( \frac{1}{2\sqrt{x}} \).

Now, applying the limit again:

\[
\lim_{{x \to 4}} \frac{a}{\frac{1}{2\sqrt{x}}} = \lim_{{x \to 4}} 2a\sqrt{x} = 4a
\]

Given that the limit equals \(-4\):

\[
4a = -4 \quad \Rightarrow \quad a = -1
\]

### Step 2: Find the value of \( b \)
We now use the fact that \( f(4) = ax + b \) when \( x = 4 \):

\[
f(4) = -1(4) + b = -4 + b
\]

From the original limit equation, we know that at \( x = 4 \), the numerator must be 0 for the limit to work, so:

\[
-4 + b = 0 \quad \Rightarrow \quad b = 4
\]

### Step 3: Calculate \( f(1) \)

Now that we know \( a = -1 \) and \( b = 4 \), substitute into the equation for \( f(x) \):

\[
f(1) = -1(1) + 4 = -1 + 4 = 3
\]

Thus, the value of \( f(1) \) is **3**.

The correct answer is **(C) 3**.

---

Would you like further details or have any questions? Here are 5 related questions to explore:

1. How do we apply L'Hopital's Rule for different types of limits?
2. What are other common indeterminate forms in calculus?
3. Can we solve this problem using a different limit technique?
4. How does the value of \( a \) influence the slope of the line \( f(x) = ax + b \)?
5. How can we determine limits involving square roots without L'Hopital's Rule?

**Tip:** L'Hopital's Rule is often useful when limits result in \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Always check for these forms before applying the rule!

Calculate f(1) given that lim_{x -> 4} (f(x)/(sqrt(x) - 2)) = -4 for f(x) = ax + b

Explore the step-by-step solution to determine the value of f(1) using L'Hopital's Rule for the function f(x) = ax + b, where the limit as x approaches 4 of f(x)/(sqrt(x) - 2) equals -4. This problem is suitable for students at the university level taking introductory Calculus courses.

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