`lim_(xto0)e^(-1//x)` is equal to :

Options

A. 0
B. `oo`
C. e
D. Does not exit

Answer

To find the limit \( \lim_{x \to 0} e^{-\frac{1}{x}} \), we will analyze the behavior of the function as \( x \) approaches 0.

### Step-by-Step Solution:

1. **Understanding the Expression**:
We start with the expression \( e^{-\frac{1}{x}} \). As \( x \) approaches 0, we need to determine what happens to \( -\frac{1}{x} \).

**Hint**: Consider the sign of \( x \) as it approaches 0 from the positive and negative sides.

2. **Approaching from the Right (Positive Side)**:
When \( x \) approaches 0 from the positive side (i.e., \( x \to 0^+ \)), \( \frac{1}{x} \) becomes very large and positive. Thus, \( -\frac{1}{x} \) approaches negative infinity.

\[ \lim_{x \to 0^+} -\frac{1}{x} = -\infty \]

**Hint**: Think about what happens to the exponential function \( e^y \) as \( y \) approaches negative infinity.

3. **Evaluating the Exponential**:
Now, substituting this back into the exponential function, we have:

\[ \lim_{x \to 0^+} e^{-\frac{1}{x}} = e^{-\infty} = 0 \]

**Hint**: Recall that \( e^{-\infty} \) represents a limit approaching zero.

4. **Approaching from the Left (Negative Side)**:
Now, consider \( x \) approaching 0 from the negative side (i.e., \( x \to 0^- \)). In this case, \( \frac{1}{x} \) becomes very large and negative, which means \( -\frac{1}{x} \) approaches positive infinity.

\[ \lim_{x \to 0^-} -\frac{1}{x} = +\infty \]

**Hint**: Again, evaluate the limit of the exponential function as the exponent approaches positive infinity.

5. **Evaluating the Exponential Again**:
Substituting this into the exponential function gives us:

\[ \lim_{x \to 0^-} e^{-\frac{1}{x}} = e^{+\infty} = \infty \]

**Hint**: Consider the implications of the limit being different from the right and left approaches.

6. **Conclusion**:
Since the limits from the right and left do not match (0 from the right and infinity from the left), the overall limit does not exist.

\[ \lim_{x \to 0} e^{-\frac{1}{x}} \text{ does not exist.} \]

### Final Answer:
The limit \( \lim_{x \to 0} e^{-\frac{1}{x}} \) does not exist.