We are tasked with finding the limit:

$\lim_{x \to 0^+} \left( 1 - e^x \right)^{\frac{1}{x}}$

Step 1: Understanding the behavior of the function as $x \to 0^+$

As $x$ approaches 0 from the right, let's first examine the behavior of $e^x$:

• $e^x \to 1$ as $x \to 0^+$.
• Therefore, $1 - e^x \to 0$ as $x \to 0^+$.

So, the expression $(1 - e^x)^{1/x}$ takes the form of $0^0$, which is an indeterminate form. Thus, we need to manipulate this limit into a solvable expression.

Step 2: Taking the natural logarithm

To simplify the expression, we take the natural logarithm of the function inside the limit:

Let:

$y = \left( 1 - e^x \right)^{\frac{1}{x}}$

Taking the natural logarithm of both sides:

$\ln(y) = \ln \left( \left( 1 - e^x \right)^{\frac{1}{x}} \right)$

Using the property of logarithms, $\ln(a^b) = b \ln(a)$, we get:

$\ln(y) = \frac{1}{x} \ln \left( 1 - e^x \right)$

Step 3: Analyzing the limit of $\ln(y)$

Now, we need to evaluate:

$\lim_{x \to 0^+} \frac{\ln(1 - e^x)}{x}$

Since $1 - e^x \to 0$ as $x \to 0^+$, we are dealing with a logarithm of a small number. To handle this, we can expand $e^x$ using its Taylor series around $x = 0$:

$e^x = 1 + x + \frac{x^2}{2} + O(x^3)$

Thus,

$1 - e^x = 1 - \left( 1 + x + \frac{x^2}{2} + O(x^3) \right) = -x - \frac{x^2}{2} + O(x^3)$

So for small $x$, we have:

$1 - e^x \approx -x$

Now, substitute this into the logarithmic expression:

$\ln(1 - e^x) \approx \ln(-x)$

For small $x$, $\ln(-x) = \ln(x) + i\pi$ (the complex logarithm), but for real values, we focus on the asymptotic behavior of $\ln(x)$.

Thus, we approximate the limit:

$\frac{\ln(1 - e^x)}{x} \approx \frac{\ln(x)}{x}$

Step 4: Evaluating the limit

Now we analyze the limit of $\frac{\ln(x)}{x}$ as $x \to 0^+$. It is known that:

$\lim_{x \to 0^+} \frac{\ln(x)}{x} = -\infty$

Step 5: Returning to the original function

Since $\ln(y) \to -\infty$, it follows that:

$y \to 0$

Thus, the original limit is:

$\lim_{x \to 0^+} \left( 1 - e^x \right)^{\frac{1}{x}} = 0$

Final Answer:

$\boxed{0}$

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Would you like a more detailed explanation or clarification on any step? Here are some related questions you could explore:

1. How do you handle limits involving indeterminate forms like $0^0$?
2. What is the Taylor series expansion of $e^x$ and how does it help in limits?
3. Can you explore the behavior of limits involving logarithms as $x \to 0$?
4. What are some common techniques for solving indeterminate limits?
5. How do you differentiate and expand logarithmic functions around small values?

Tip: When encountering limits of the form $0^0$, consider using logarithms to transform the problem into a more manageable form.