https://file.aiquickdraw.com/mathbot/file/1726320913252pybzra7x.jpg

The expression provided is:

\[
\lim_{x \to 0} \frac{\sqrt{1 + x} - 1}{x}
\]

### Step-by-step Solution:

This is a standard limit problem, and we can solve it using **L'Hôpital's Rule** or by simplifying the expression.

1. **Direct Substitution**: If we substitute \(x = 0\) into the expression, we get:
   \[
   \frac{\sqrt{1 + 0} - 1}{0} = \frac{1 - 1}{0} = \frac{0}{0}
   \]
   This is an indeterminate form \( \frac{0}{0} \), so we can apply **L'Hôpital's Rule**, which involves differentiating the numerator and denominator.

2. **Applying L'Hôpital's Rule**:
   - Differentiate the numerator: \( \frac{d}{dx} \left( \sqrt{1 + x} - 1 \right) = \frac{1}{2\sqrt{1 + x}} \)
   - Differentiate the denominator: \( \frac{d}{dx}(x) = 1 \)

   Now, the limit becomes:
   \[
   \lim_{x \to 0} \frac{\frac{1}{2\sqrt{1 + x}}}{1} = \lim_{x \to 0} \frac{1}{2\sqrt{1 + x}} = \frac{1}{2}
   \]

### Final Answer:
\[
\lim_{x \to 0} \frac{\sqrt{1 + x} - 1}{x} = \frac{1}{2}
\]

Would you like any further details or clarifications?

### Follow-up Questions:
1. What is L'Hôpital's Rule, and when can we apply it?
2. How do we simplify expressions involving square roots in limits?
3. Can this problem be solved using a different method, such as multiplying by a conjugate?
4. Why does direct substitution result in an indeterminate form here?
5. How do we differentiate functions involving square roots?

### Tip:
When facing limits involving square roots, a common strategy is to multiply the numerator and denominator by the conjugate to simplify the expression before applying L'Hôpital's Rule.

Find the limit of (sqrt(1 + x) - 1) / x as x approaches 0.

This problem involves finding the limit of (sqrt(1 + x) - 1) / x as x approaches 0. The solution includes applying L'Hopital's Rule due to the indeterminate form 0/0. Step-by-step guidance is provided for differentiating the numerator and denominator.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation