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To evaluate the limit:

\[
\lim_{{x \to 2}} \frac{{f(x) - 3}}{{x - 2}}
\]

we need to determine the behavior of \( f(x) \) around \( x = 2 \).

1. From the graph, the value \( f(2) = 3 \).
2. The limit resembles the definition of the derivative of \( f(x) \) at \( x = 2 \), as it has the form \(\frac{{f(x) - f(2)}}{{x - 2}}\).
3. To find this derivative, examine the slope of the line segment near \( x = 2 \). From \( x = 1 \) to \( x = 3 \), \( f(x) \) increases from 2 to 4, yielding a slope of \( \frac{4 - 2}{3 - 1} = 1 \).

Thus, the limit is:

\[
\lim_{{x \to 2}} \frac{{f(x) - 3}}{{x - 2}} = 1
\]

Would you like a deeper explanation on finding derivatives from graphs?

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1. What does a limit represent in calculus?
2. How can we interpret derivatives graphically?
3. What happens if a function is not continuous at the limit point?
4. Can you find the derivative at other points of the graph?
5. How is this limit form related to the concept of tangent lines?

**Tip:** When you see limits in the form \( \frac{{f(x) - f(c)}}{{x - c}} \), think of derivatives.

Evaluate \( \lim_{{x \to 2}} \frac{{f(x) - 3}}{{x - 2}} \) based on the given graph of \( f(x) \).

Learn how to evaluate the limit \( \lim_{{x \to 2}} \frac{{f(x) - 3}}{{x - 2}} \) by analyzing the slope on a given graph of \( f(x) \). This problem explores the relationship between limits and derivatives, suitable for students in Grades 11-12.

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