https://file.aiquickdraw.com/mathbot/file/1726030604037vt47czk7.jpg

The limit in the image is:

\[
\lim_{x \to 0} \frac{\sqrt{1 + x} \cdot \sqrt{1 + 2x} \cdot \sqrt{1 + 3x} \cdot \ldots \cdot \sqrt{1 + nx - 1}}{x}
\]

We need to evaluate this limit as \( x \to 0 \).

### Step-by-step solution:

1. **Expression Simplification:**
   The expression inside the limit involves a product of square roots. Expanding each term using the first-order approximation of a binomial expansion for small \( x \), we have:
   \[
   \sqrt{1 + kx} \approx 1 + \frac{kx}{2}
   \]
   for each \( k = 1, 2, 3, \ldots, n \). Thus, the product of all these terms becomes:
   \[
   \prod_{k=1}^{n} \sqrt{1 + kx} \approx \prod_{k=1}^{n} \left( 1 + \frac{kx}{2} \right)
   \]
   When expanded, this product can be approximated as:
   \[
   \prod_{k=1}^{n} \left( 1 + \frac{kx}{2} \right) \approx 1 + \frac{1x}{2} + \frac{2x}{2} + \frac{3x}{2} + \ldots + \frac{(n-1)x}{2}
   \]
   which simplifies to:
   \[
   1 + \frac{x}{2} \sum_{k=1}^{n-1} k = 1 + \frac{x}{2} \cdot \frac{(n-1)n}{2}
   \]
   Therefore, the approximation becomes:
   \[
   1 + \frac{(n-1)nx}{4}
   \]

2. **Simplifying the Limit:**
   Now, using this approximation in the original limit:
   \[
   \lim_{x \to 0} \frac{1 + \frac{(n-1)nx}{4} - 1}{x}
   \]
   Simplifies to:
   \[
   \lim_{x \to 0} \frac{\frac{(n-1)nx}{4}}{x} = \frac{(n-1)n}{4}
   \]

Thus, the value of the limit is:

\[
\boxed{\frac{(n-1)n}{4}}
\]

---

Do you need more details on any step? Here are some related questions:

1. What is the general form of a binomial expansion for small values of \(x\)?
2. How do we handle products of approximations in limits?
3. How does the binomial approximation simplify square roots?
4. What would change if we evaluated this limit for specific small values of \(n\)?
5. How does L'Hopital's rule apply to limits involving \(0/0\) forms?

**Tip:** When working with limits involving products or sums, it is often useful to apply approximations for small values of \(x\), like the binomial expansion, to simplify the expression.

Evaluate the limit: lim_{x \to 0} \frac{\sqrt{1 + x} \cdot \sqrt{1 + 2x} \cdot \sqrt{1 + 3x} \cdot \ldots \cdot \sqrt{1 + nx - 1}}{x}

This page explains the step-by-step solution to evaluate the limit lim_{x \to 0} \frac{\sqrt{1 + x} \cdot \sqrt{1 + 2x} \cdot \sqrt{1 + 3x} \cdot \ldots \cdot \sqrt{1 + nx - 1}}{x}. The solution involves concepts like binomial expansion, series approximation, and first-order approximation, suitable for university-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation