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Solve example 1 in detail with explanation 

The image provided contains Example 1, which appears to be focused on solving limits.

Let's walk through the detailed solution process step by step:

### Problem
We are asked to find the limit:

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

### Step-by-step Solution

1. **Recognize the Standard Limit Form**: 
   The expression resembles a well-known standard limit form:
   \[
   \lim_{{x \to 0}} \frac{e^x - 1}{x} = 1
   \]
   This result comes from the derivative of the exponential function at 0. Specifically, the derivative of \( e^x \) at \( x = 0 \) is 1. So we can directly apply this standard limit formula.

2. **Application of L'Hopital's Rule (if necessary)**:
   If you were not familiar with the standard form, you could solve it using **L'Hopital's Rule**. L'Hopital's Rule applies to limits of indeterminate forms like \( \frac{0}{0} \). To apply the rule, take the derivatives of the numerator and denominator:

   - The derivative of \( e^x - 1 \) is \( e^x \)
   - The derivative of \( x \) is 1.

   So applying L'Hopital's Rule gives us:
   \[
   \lim_{{x \to 0}} \frac{e^x - 1}{x} = \lim_{{x \to 0}} \frac{e^x}{1} = e^0 = 1
   \]

3. **Conclusion**: 
   The limit evaluates to \( 1 \), either by applying the standard result or by using L'Hopital's Rule.

\[
\boxed{1}
\]

---

This is the detailed explanation of the example in your image. If you'd like to delve further into limits or related topics, feel free to ask!

### Additional Relative Questions:
1. What are other standard limit forms similar to the one used here?
2. How does L'Hopital's Rule apply in more complicated limit problems?
3. Can the limit \( \lim_{{x \to 0}} \frac{e^x - 1}{x} \) be solved using a series expansion of \( e^x \)?
4. How do limits involving trigonometric functions differ from exponential functions?
5. What role does continuity play in determining limits of functions like \( e^x \)?

**Tip:** When dealing with limits, knowing key standard forms like \( \lim_{{x \to 0}} \frac{\sin x}{x} = 1 \) and \( \lim_{{x \to 0}} \frac{e^x - 1}{x} = 1 \) can save you time in computations.

Solve example 1 in detail with explanation.

Solve the limit problem lim_{x -> 0} (e^x - 1)/x using standard limit forms and L'Hopital's Rule. This detailed explanation walks through the key steps and concepts involved, suitable for students in Grades 11-12 or undergraduate levels.

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