https://file.aiquickdraw.com/mathbot/file/1728015779985615clzym.jpg

The problem presented in the image states the following:

If 

\[
\lim_{{x \to 3}} \frac{g(3) - g(x)}{3 - x} = -0.628,
\]

then near the point where \( x = 3 \), the graph of \( g(x) \):

(A) is decreasing  
(B) is increasing  
(C) is concave downwards  
(D) has a point of inflection

### Solution:

The limit expression

\[
\lim_{{x \to 3}} \frac{g(3) - g(x)}{3 - x}
\]

is the **definition of the derivative** of \( g(x) \) at \( x = 3 \), except with a negative sign due to the order of the terms. Normally, the derivative is written as:

\[
g'(3) = \lim_{{x \to 3}} \frac{g(x) - g(3)}{x - 3}.
\]

The expression given in the problem is essentially \( -g'(3) \), so we conclude:

\[
-g'(3) = -0.628 \quad \text{or} \quad g'(3) = 0.628.
\]

Since \( g'(3) = 0.628 \), which is positive, the function \( g(x) \) is **increasing** at \( x = 3 \).

Thus, the correct answer is:

\[
\boxed{B}.
\]

### Would you like further explanation or have any other questions?

### Here are 5 related questions to deepen your understanding:

1. What is the geometric interpretation of the derivative of a function at a point?
2. How does the sign of the first derivative influence the increasing or decreasing nature of a function?
3. What are the conditions for concavity, and how is the second derivative used to determine concavity?
4. Can a function be increasing and concave down at the same point?
5. How can you determine if a point is a point of inflection?

#### Tip: 
Always remember that the first derivative provides information about the slope of the graph, while the second derivative tells you about the concavity.

If lim_{x → 3} (g(3) - g(x)) / (3 - x) = -0.628, then near the point where x = 3, the graph of g(x): (A) is decreasing, (B) is increasing, (C) is concave downwards, (D) has a point of inflection.

Explore how to determine the behavior of a function using the limit definition of a derivative. This calculus problem focuses on using limits to find g'(x) at x = 3 and determining whether the function is increasing or decreasing based on the result.

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