Answer is: option4

Solution:

The left-hand limit \(\lim\limits_{x \to a^-} f(x)\) and the right-hand limit \(\lim\limits_{x \to a^+} f(x)\) must be equal for the limit to exist.

Observing the graph, both sides of \(x = a\) approach the same value.

Hence, the limit exists, so Statement I is true.

The function \( f(x) \) is defined at \( x = a \) if there is a solid dot at \( x = a \).

From the graph, at \( x = a \), there is an open circle (indicating the function is not defined there), but a filled point is present below it at a different value.

This means \( f(a) \) is defined, so Statement II is true.

A relative minimum occurs where \( f(a) \) is less than the values of \( f(x) \) for nearby \( x \).

The graph shows that \( f(a) \) is below the neighboring function values.

This confirms \( f(x) \) has a relative minimum at \( x = a \).

Hence, Statement III is true.

Since all three statements are correct, the correct answer is: I, II, and III.