The problem asks to find $\lim_{{x \to -5}} f(x)$ using the graph provided.

To determine this limit, we need to check the behavior of the function $f(x)$ as $x$ approaches $-5$ from both the left and the right:

1. From the left side ($x \to -5^-$): As $x$ approaches $-5$ from values less than $-5$, the graph shows that $f(x)$ is approaching a value around $y = 5$.

2. From the right side ($x \to -5^+$): As $x$ approaches $-5$ from values greater than $-5$, the graph shows that $f(x)$ also approaches the same value, $y = 5$.

Since both the left-hand and right-hand limits are equal and approach $5$, the overall limit is: $\lim_{{x \to -5}} f(x) = 5$

Would you like more details or explanations about this solution?

Here are 5 related questions to deepen your understanding:

1. How do you determine left-hand and right-hand limits from a graph?
2. What does it mean if the left-hand limit and right-hand limit are different at a point?
3. What if the function had a hole or jump discontinuity at $x = -5$? How would that affect the limit?
4. Can a limit exist at a point where the function is not defined?
5. How does one deal with vertical asymptotes when finding limits?

Tip: When analyzing limits graphically, always check both directions to ensure the function approaches the same value from the left and right!