Read the following information carefully and answer the question given below:
Consider the function `f(x)={{:(x^(2)-5,xle3),(sqrt(x+13),xgt3):}`
Consider the following statements:
I. The function is discontinuous at x = 3
II. The function is not differentiable at x = 0
Which of the above statement(s) is/are correct?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To solve the problem, we need to analyze the function \( f(x) \) given by:

\[ f(x) = \begin{cases} x^2 - 5 & \text{if } x \leq 3 \\ \sqrt{x + 13} & \text{if } x > 3 \end{cases} \]

We need to evaluate the two statements provided:

### Statement I: The function is discontinuous at \( x = 3 \)

1. **Find the left-hand limit as \( x \) approaches 3:**
\[ \lim_{x \to 3^-} f(x) = \lim_{x \to 3} (x^2 - 5) = 3^2 - 5 = 9 - 5 = 4 \]

2. **Find the right-hand limit as \( x \) approaches 3:**
\[ \lim_{x \to 3^+} f(x) = \lim_{x \to 3} \sqrt{x + 13} = \sqrt{3 + 13} = \sqrt{16} = 4 \]

3. **Check the value of the function at \( x = 3 \):**
\[ f(3) = 3^2 - 5 = 4 \]

4. **Conclusion for Statement I:**
Since the left-hand limit, right-hand limit, and the value of the function at \( x = 3 \) are all equal to 4, the function is continuous at \( x = 3 \). Therefore, Statement I is **incorrect**.

### Statement II: The function is not differentiable at \( x = 0 \)

1. **Check the differentiability at \( x = 0 \):**
The function \( f(x) = x^2 - 5 \) for \( x \leq 3 \) is a polynomial function, which is differentiable everywhere in its domain, including at \( x = 0 \).

2. **Conclusion for Statement II:**
Since \( f(x) \) is differentiable at \( x = 0 \), Statement II is also **incorrect**.

### Final Conclusion:
Both statements are incorrect. Therefore, the answer is that neither statement 1 nor statement 2 is correct.

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