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The problem asks to evaluate the following limit:

\[
\lim_{{x \to 0^-}} \frac{2}{1 - e^{1/x}}
\]

Let's analyze the behavior of this limit step by step:

### Step 1: Behavior of \( e^{1/x} \) as \( x \to 0^- \)

- When \( x \) approaches \( 0 \) from the left (i.e., \( x \to 0^- \)), the term \( \frac{1}{x} \) becomes a large negative number. Therefore, \( e^{1/x} \) approaches \( 0 \), because the exponential of a large negative number tends to zero:
  \[
  e^{1/x} \to 0 \quad \text{as} \quad x \to 0^-.
  \]

### Step 2: Evaluate the expression \( 1 - e^{1/x} \)

- As \( e^{1/x} \to 0 \), we have:
  \[
  1 - e^{1/x} \to 1 \quad \text{as} \quad x \to 0^-.
  \]

### Step 3: Evaluate the limit

- The limit now becomes:
  \[
  \lim_{{x \to 0^-}} \frac{2}{1 - e^{1/x}} = \frac{2}{1} = 2.
  \]

Thus, the limit evaluates to:

\[
\boxed{2}
\]

Would you like more details or have any questions?

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Here are five related questions to expand your understanding:
1. What happens to \( e^{1/x} \) when \( x \to 0^+ \) instead of \( x \to 0^- \)?
2. How would the limit change if the numerator were \( x \) instead of a constant?
3. Can you generalize the behavior of \( e^{f(x)} \) for different functions \( f(x) \)?
4. What if the limit had a denominator of \( 1 + e^{1/x} \) instead of \( 1 - e^{1/x} \)?
5. What are the possible strategies for dealing with indeterminate forms in limits?

### Tip:
When analyzing limits, always pay close attention to the behavior of functions in the limit's neighborhood (whether approaching from the left or right). This can help identify the correct limit behavior in different cases.

Evaluate the limit lim_{x -> 0^-} 2 / (1 - e^(1/x))

Explore how to calculate the limit of 2 / (1 - e^(1/x)) as x approaches 0 from the left. This problem involves analyzing the behavior of exponential functions in limits and is suitable for undergraduate calculus students.

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