Consider the following statements :
I . `lim_(xto0)(1)/(x)` exists .
II `lim_(xto0)e^(1//x)` does not exists Which of the above statements is /are correct ?

Options

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

Answer

To solve the given problem, we will analyze both statements separately.

### Statement I:
**Evaluate** \( \lim_{x \to 0} \frac{1}{x} \)

1. **Substituting the limit**:
When we directly substitute \( x = 0 \) into \( \frac{1}{x} \), we get \( \frac{1}{0} \), which is undefined.

2. **Analyzing the limit from both sides**:
- As \( x \) approaches 0 from the positive side (i.e., \( x \to 0^+ \)), \( \frac{1}{x} \) approaches \( +\infty \).
- As \( x \) approaches 0 from the negative side (i.e., \( x \to 0^- \)), \( \frac{1}{x} \) approaches \( -\infty \).

3. **Conclusion**:
Since the left-hand limit and the right-hand limit do not agree (one approaches \( +\infty \) and the other \( -\infty \)), the limit \( \lim_{x \to 0} \frac{1}{x} \) does not exist.

### Statement II:
**Evaluate** \( \lim_{x \to 0} e^{\frac{1}{x}} \)

1. **Substituting the limit**:
When we substitute \( x = 0 \), we get \( e^{\frac{1}{0}} \), which is \( e^{\infty} \) or \( e^{-\infty} \) depending on the direction from which we approach 0.

2. **Analyzing the limit from both sides**:
- As \( x \) approaches 0 from the positive side (i.e., \( x \to 0^+ \)), \( \frac{1}{x} \) approaches \( +\infty \), thus \( e^{\frac{1}{x}} \) approaches \( e^{+\infty} = +\infty \).
- As \( x \) approaches 0 from the negative side (i.e., \( x \to 0^- \)), \( \frac{1}{x} \) approaches \( -\infty \), thus \( e^{\frac{1}{x}} \) approaches \( e^{-\infty} = 0 \).

3. **Conclusion**:
Since the left-hand limit and the right-hand limit do not agree (one approaches \( +\infty \) and the other \( 0 \)), the limit \( \lim_{x \to 0} e^{\frac{1}{x}} \) does not exist.

### Final Conclusion:
- Statement I is **incorrect**: \( \lim_{x \to 0} \frac{1}{x} \) does not exist.
- Statement II is **correct**: \( \lim_{x \to 0} e^{\frac{1}{x}} \) does not exist.

### Answer:
Only Statement II is correct.

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