From the graph you provided, we can analyze the limits and function values at various points. Here's the detailed assessment:

a) $\lim_{x \to -5} f(x)$

• As $x$ approaches -5 from either side, the y-value approaches 10.
• Answer: 10

b) $\lim_{x \to -5^-} f(x)$

• As $x$ approaches -5 from the left, the y-value approaches 10.
• Answer: 10

c) $\lim_{x \to -5^+} f(x)$

• As $x$ approaches -5 from the right, the y-value approaches 10.
• Answer: 10

d) $f(-5)$

• The value of the function at $x = -5$ is 10 (filled circle on the graph indicates the function value at this point).
• Answer: 10

e) $\lim_{x \to 0} f(x)$

• As $x$ approaches 0, the y-value approaches approximately 0 (following the curve to where it crosses the y-axis).
• Answer: 0

f) $f(0)$

• The value of the function at $x = 0$ is 0, indicated by the curve crossing the y-axis at this point.
• Answer: 0

g) $\lim_{x \to -5^-} f(x)$

• This is a repeat of part (b) and the answer remains 10.
• Answer: 10

h) $\lim_{x \to -5^+} f(x)$

• This is a repeat of part (c) and the answer remains 10.
• Answer: 10

i) $\lim_{x \to 5} f(x)$

• As $x$ approaches 5, the function value doesn’t appear to stabilize due to the graph tending steeply downwards without leveling off, suggesting that the limit does not exist (DNE).
• Answer: DNE

j) $f(5)$

• At $x = 5$, there is a hollow circle on the graph indicating that the function value is not defined at this point.
• Answer: DNE

If you need further details or have any questions, please let me know.

Here are five related questions you might consider:

1. What happens to the limit of $f(x)$ as $x$ approaches a point where the function has a vertical asymptote?
2. How would you determine the limit of $f(x)$ as $x$ approaches infinity given a different function graph?
3. Can a function have limits at points where it is not continuous?
4. What does a hollow circle on a function graph typically indicate about the function value at that point?
5. How can the behavior of $f(x)$ near $x = -10$ and $x = 10$ help us understand the end behavior of the function?

Tip: Remember that limits can exist even at points where the function itself is not defined, indicated by hollow circles on the graph. Always check the behavior of the graph approaching these points.