Consider the following statements in respect of the function `f(x)=sin((1)/(x))` for `xne0 and f(0)=0`
1. `underset(xrarr0)lim f(x)` exists
2. f(x) is continuous at x = 0
Which of the above statements is/are correct?

Options

A. A) 1 only
B. B) 2 only
C. C) Both 1 and 2
D. D) Neither 1 nor 2

Answer

To analyze the function \( f(x) = \sin\left(\frac{1}{x}\right) \) for \( x \neq 0 \) and \( f(0) = 0 \), we need to evaluate the two statements given in the question.

### Step 1: Evaluate the limit as \( x \) approaches 0
We need to find \( \lim_{x \to 0} f(x) \).

\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \sin\left(\frac{1}{x}\right) \]

As \( x \) approaches 0, \( \frac{1}{x} \) approaches infinity. The sine function oscillates between -1 and 1 for all real numbers. Therefore, as \( x \) approaches 0, \( \sin\left(\frac{1}{x}\right) \) does not approach a single value but continues to oscillate.

### Conclusion for Statement 1
Since the limit does not settle to a specific value, we conclude that:

\[ \lim_{x \to 0} f(x) \text{ does not exist.} \]

Thus, Statement 1 is **incorrect**.

### Step 2: Check continuity at \( x = 0 \)
A function is continuous at a point if the following three conditions are met:

1. \( f(c) \) is defined.
2. \( \lim_{x \to c} f(x) \) exists.
3. \( \lim_{x \to c} f(x) = f(c) \).

Here, we have \( c = 0 \).

- \( f(0) = 0 \) is defined.
- We found that \( \lim_{x \to 0} f(x) \) does not exist.

Since the limit does not exist, the second condition for continuity fails. Therefore, \( f(x) \) is not continuous at \( x = 0 \).

### Conclusion for Statement 2
Since the limit does not exist, we conclude that:

\[ f(x) \text{ is not continuous at } x = 0. \]

Thus, Statement 2 is also **incorrect**.

### Final Answer
Both statements are incorrect.

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