Let's revisit the problem to clarify where the correct limit of $2$ comes from.

Re-analysis of the Graph Near $x = 4$:

1. Right-Hand Limit Definition: The right-hand limit $\lim_{{x \to 4^+}} f(x)$ looks at the value that $f(x)$ approaches as $x$ approaches 4 from values greater than 4.

2. Observing the Graph:

   • When approaching $x = 4$ from the right (where $x$ is slightly greater than 4), trace along the graph towards $x = 4$.
   • As you do this, the graph shows that the value of $f(x)$ is approaching 2 from above.

Conclusion:

• This corrects the earlier observation: as $x$ approaches 4 from the right, $f(x)$ indeed approaches 2, not 3.

Thus, the correct answer is:

$\lim_{{x \to 4^+}} f(x) = 2$

I apologize for the earlier mistake. Thanks for catching that! Would you like to go through more similar problems to solidify your understanding, or is there something else you'd like to explore?

Further Questions for Practice:

1. What is $\lim_{{x \to 4^-}} f(x)$ based on the graph?
2. Determine if $f(x)$ is continuous at $x = 4$.
3. What is the value of $f(4)$?
4. How does $f(x)$ behave as $x$ approaches 0?
5. Find the left-hand and right-hand limits at $x = -2$.

Tip:

When analyzing graphs, take note of any jumps, breaks, or asymptotes at key points, as these features often indicate where the limits may differ or where discontinuities occur.