Based on the description of the graph, I'll go through each statement one by one and determine whether it is true or false.

Statement a:

"The statement $\lim_{{x \to 0}} f(x)$ exists is true."

• This statement refers to whether the limit of $f(x)$ as $x$ approaches 0 exists. A limit exists if the function approaches the same value from both the left and the right sides as $x \to 0$.
• From the description, we know that at $x = 0$, there is a point at $(0, 1)$, but the behavior of the graph near $x = 0$ is not fully specified. If the graph approaches the same value from both sides, the limit exists. Since there's no information about a discontinuity at $x = 0$, we assume the limit exists.

Conclusion: True (the limit exists, based on the provided description).

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Statement b:

"The statement $\lim_{{x \to 0}} f(x) = 0$ is true."

• The limit as $x \to 0$ would equal 0 if the function approaches the value 0 from both sides of $x = 0$. However, it is given that a point is plotted at $(0, 1)$, which means the function value at $x = 0$ is 1, not 0.
• Since the function is continuous at $x = 0$ (i.e., the value at $(0, 1)$ is the limit from both sides), the limit does not equal 0.

Conclusion: False (the limit is not 0).

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Statement c:

"The statement $\lim_{{x \to 0}} f(x) = 1$ is true."

• Given that the graph passes through the point $(0, 1)$, we can infer that the function value at $x = 0$ is 1.
• If the function approaches the value of 1 from both sides as $x$ approaches 0, then the limit as $x \to 0$ would indeed be 1.

Conclusion: True (the limit equals 1, based on the provided description).

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Summary of answers:

• a. True: $\lim_{{x \to 0}} f(x)$ exists.
• b. False: $\lim_{{x \to 0}} f(x) = 0$.
• c. True: $\lim_{{x \to 0}} f(x) = 1$.

Would you like to dive deeper into any of these statements, or do you have further questions? Here are some related questions you might find interesting:

1. What does it mean for a limit to "exist" at a specific point?
2. How do we determine if a function is continuous at a point?
3. How does an open or closed circle on a graph affect the limit of a function?
4. What are some examples of limits that do not exist?
5. What is the significance of the behavior of a function near discontinuities?

Tip: When evaluating limits, always check both the left-hand limit ($\lim_{{x \to c^-}}$) and right-hand limit ($\lim_{{x \to c^+}}$) separately. The limit exists only if both one-sided limits agree.