Consider the following :
1. `x+x^(2)` is continuous at x = 0
2. `x+cos""(1)/(x)` is discontinuous at x = 0
3. `x^(2)+cos""(1)/(x)` is continuous at x = 0
Which of the above are correct?

Options

A. 1 and 2
B. 2 and 3
C. 1 and 3
D. 2, 2 and 3

Answer

To determine the correctness of the statements regarding continuity at \( x = 0 \), we will analyze each statement step by step.

### Step 1: Analyze the first statement \( f(x) = x + x^2 \)

1. **Check continuity at \( x = 0 \)**:
- We need to check if \( \lim_{x \to 0} f(x) = f(0) \).
- Calculate \( f(0) \):
\[ f(0) = 0 + 0^2 = 0 \]
- Now, calculate the limit:
\[ \lim_{x \to 0} (x + x^2) = \lim_{x \to 0} x + \lim_{x \to 0} x^2 = 0 + 0 = 0 \]
- Since \( \lim_{x \to 0} f(x) = f(0) = 0 \), the function is continuous at \( x = 0 \).

### Conclusion for Statement 1:
- The first statement is **true**.

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### Step 2: Analyze the second statement \( f(x) = x + \frac{\cos(1)}{x} \)

1. **Check continuity at \( x = 0 \)**:
- We need to check if \( \lim_{x \to 0} f(x) = f(0) \). However, \( f(0) \) is not defined because \( \frac{\cos(1)}{0} \) is undefined.
- Calculate the limit:
\[ \lim_{x \to 0} \left( x + \frac{\cos(1)}{x} \right) \]
- As \( x \) approaches 0, \( \frac{\cos(1)}{x} \) approaches \( \infty \) (or \( -\infty \) depending on the direction of approach), making the limit undefined.
- Since \( f(0) \) is undefined and the limit does not exist, the function is discontinuous at \( x = 0 \).

### Conclusion for Statement 2:
- The second statement is **true**.

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### Step 3: Analyze the third statement \( f(x) = x^2 + \frac{\cos(1)}{x} \)

1. **Check continuity at \( x = 0 \)**:
- Similar to the previous case, we need to check if \( \lim_{x \to 0} f(x) = f(0) \). Again, \( f(0) \) is not defined because \( \frac{\cos(1)}{0} \) is undefined.
- Calculate the limit:
\[ \lim_{x \to 0} \left( x^2 + \frac{\cos(1)}{x} \right) \]
- As \( x \) approaches 0, \( x^2 \) approaches 0, but \( \frac{\cos(1)}{x} \) approaches \( \infty \) (or \( -\infty \)), making the limit undefined.
- Since \( f(0) \) is undefined and the limit does not exist, the function is discontinuous at \( x = 0 \).

### Conclusion for Statement 3:
- The third statement is **false**.

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### Final Conclusion:
- The correct statements are **1 and 2**. Therefore, the answer is that statements 1 and 2 are true, while statement 3 is false.

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